Next: Contraction Mapping Up: SOLUTION OF NONLINEAR EQUATIONS Previous: Secant Method

## Fixed Point Iteration

def: Fixed Point is the value such that      .

Fixed Point problems and root-finding problems are equivalent: let . Hence, if a function has a fixed point then has a root.

Three Problems:

1. What functions have a fixed point?
2. How do we determine the fixed point?
3. Is the fixed point unique?
Example      has a fixed point at each ....just plot.

Example has 2 fixed points in     ....plot, again.

Theorem (Existence and Uniqueness): If and then has a fixed point in .

Suppose, in addition, that exists on and that a positive constant exists with

then the fixed point in is unique.

Proof:

If or , then existence of fixed point is clear. Suppose not, then it must be true that and . Define . Then is continuous on and

The Intermediate Value Theorem implies that there exist for which . Thus, is fixed point of .

holds and that and are both fixed points in with . By the Mean Value Theorem a number exists between and such that

This contradiction comes from and the fixed point is unique.

Fixed-Point Iteration

Pick a and generate a sequence such that If the sequence converges to and is continuous then by the theorem above:

The algorithm is depicted in Figure 18

Fixed Point Algorithm

Input: TOL,
Output: or message of failure
Step 1: Set
Step 2: While do Steps 3 - 6
Step 3: Set     % Compute
Step 4: if TOL then
output ;     % found not
Stop
Step 5: Set
Step 6: Set     Update .
Step 7: Output (Iterations exceeded. )
END

Theorem: (Fixed Point Iteration) Let and suppose . Suppose in addition that is continuous on with

if then for any in , the sequence

converges only linearly to the unique fixed point in

Proof: Fixed Point Theorem says . Since exists on apply mean value Theorem to to show that for any

since , and . Since is continuous on we have

thus

Remark: Can get higher-order convergence when

Theorem: Suppose solution of . and continuous and strictly bounded by on an open interval containing . Then such that . The sequence converges quadratically:

Fixed Point Iteration: let and suppose . Suppose in addition that exists on with

 (6)

if is any number in , then

converges to unique fixed point in .

Proof: From fixed point theorem, a unique fixed point exists in . Since maps into itself, the sequence is defined and .

Using (6) and the Mean Value Theorem

By induction
 (7)

Since

and

Corollary If satisfies hypothesis of Fixed Point Iteration theorem, a bounds for the error involved in using to approximate are given by

Proof: (a) follows from (7):

For

therefore for

Since then

Since then

Remark: Rate depends on . The smaller , the faster it converges.

Example: Consider for . This function is illustrated in Figure 19. Get matlab code used in the example.

First we wish to ensure that the function maps into itself.

Next we look at the derivative of

This fulfills the requirements for a unique fixed point to exist in . It also ensures that if we start with any non-negative value we will converge to the fixed point. The table below shows the first ten iterations for three different values of . Figure 20a and Figure 20b illustrate the iteration history and the logarithm of the error, for a case starting with .

Figure 21a and Figure 21b illustrate the iteration history and the logarithm of the error, for a case starting with .
Finally, Figure 22a and Figure 22b illustrate the iteration history and the logarithm of the error, for a case starting with .

The iterations for the three different starting points all appear to converge on . The log error plots are straight lines and they all have the same slope, indicating the same rate of convergence.

We can also prove analytically that is the fixed point of . A fixed point of satisfies

We can rearrange this to get which has one real root, .

Example: Consider the function for . We will start with the initial value and consider what happens for various values of . -Figure show the iterations for , respectively.

They also plot on the same graph as so we can see the fixed point, and finally plot the log of the error for each value of . We can see that for both and the iteration converges to a fixed point. The log error plots are straight lines with the slope showing the convergence rate (Question: Why does the log error plot flatten off for ?). In the case we can see that it converges faster than the case . For the fixed point does not converge but seems to bounce around different values in the interval . In fact for values of between and we get all sorts of interesting beheviour. For more information on this click here. But when we make less than 0.5 the iteration is able to escape from the interval and once it does this it increases rapidly with each iteration until it causes an overflow error in the computer.

(b) . (c) . (d) .">Get matlab code used in the example.

Now lets see whether we can understand what is happening. First let us look at the range of the function

This shows why the iterations blow up for less than 0.5. For the range is not within the domain of (i.e. ) and so points may 'escape'. However for any value of greater than a half the range is mapped to within the domain.

Next we need to look at the derivative of

The magnitude of the derivative is only less than one for all values of if . Thus for any value of greater than two the fixed point theorem holds and we have guaranteed convergence. We know, however, that we still get convergence to a fixed point for some values of less than two. What is happening in these cases?

If the magnitude of the derivative will be less than one for . As long as the fixed point lies within this interval the theorem tell us that there will be a region around the fixed point where iterations will converge to the fixed point. This is the case as long as . As it turns out, we may still start at any point within and we will eventually arrive at the fixed point although convergence takes longer and longer the closer is to the critical point.

For values of the fixed point still exists but it becomes unstable (i.e. If you start close to the fixed point and iterate you will move away from it rather than towards it).

If we plot and the line on the same graph we can see that there is only one fixed point within the interval for all values of . In fact we can calculate the value of the fixed point analytically by solving .

This is a simple quadratic equation with two solutions

For only the smaller of the two solutions lies within the interval and is the unique fixed point.

Subsections

Next: Contraction Mapping Up: SOLUTION OF NONLINEAR EQUATIONS Previous: Secant Method
Juan Restrepo 2003-04-12


Math

What substitution could I use to integrate

a/(a^2 + x^2)^3/2 dx

1. 👍
2. 👎
3. 👁
1. Let u = x/(x^2 + a^2)^1/2

and you will find that

(1/a^2)* du
= integral of dx/(x^2+a^2)^3/2
which is the integral you want.
Therefore u/a^2
= (x/a^2)/(x^2 + a^2)^1/2

1. 👍
2. 👎
2. Computer program says the answer is

x/(a*(a^2 + x^2)^(1/2))

which is slightly different from your answer. Thanks so much for the help on this one. I was really stuck.

1. 👍
2. 👎
3. I did x = a tan u
dx = a (sec u)^2 du

int of a/(a^2 + x^2)^3/2 dx
= int of (a sec u)^2/(a^2 + (a tan u)^2)^3/2 du
= int of (a sec u)^2/(a sec u)^3 du
= int of (cos u)/a du
= (sin u)/a + K
since u = atan (x/a)
= x/(a*(a^2 + x^2)^(1/2)) + K

Thanks again...

1. 👍
2. 👎

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# All invariant subspaces of a linear operator $T$

Let $T \in \mathscr L(\mathbb F^n)$ such that $T(x_1,x_2,...,x_n)=(x_1,2x_2,...,nx_n)$. Then find all the invariant subspaces of $T$. Clearly, $Null$ $T$ and $Range$ $T$ are two invariant subspaces.

Also, all the subspaces spanned by the eigen vectors form $1$-dimensional invariant subspaces. In this case the eigen values of $T$ are $i$, where $i$$\in\{1,2,...,n\} and the corresponding eigen vector is of form (0,...,0,a,0,...,0) where a(\neq 0)\in \mathbb F is the i^{th}-component of the vector. But what about the invariant subspaces of other dimensions? For instance, if W is an invariant subspace of T of dimension k, then T|_W is a linear operator on W. So, for any w\in W, Tw\in W. If w\in span(e_1,e_2,...,e_k) then Tw\in span(e_1,2e_2,...,ke_k). Since, dim W= dim Tw=k, we can say that Tw=span(e_1,2e_2,...,ke_k)\in span(e_1,e_2,...,e_k)=W i.e Tw\in W. So, can we conclude from here that there will be invariant subspaces of all dimensions under T, given by span(e_1,e_2,...,e_k), where 1\leqslant k\leqslant n (which are precisely 2^n in number)? • What do you think the invariant subspaces are? – Demophilus Oct 22 '17 at 13:32 • @Demophilus If W is an invariant subspace of T of dimension k, then T|_W is a linear operator on W. So, for any w\in W, Tw\in W. If w\in span(e_1,e_2,...,e_k) then Tw\in span(e_1,2e_2,...,ke_k). Since, dim W= dim Tw=k, we can say that Tw=span(e_1,2e_2,...,ke_k)\in span(e_1,e_2,...,e_k)=W i.e Tw\in W. So, can we conclude from here that there will be invariant subspaces of all dimensions under T, given by span(e_1,e_2,...,e_k), where 1\leqslant k\leqslant n? – JackT Oct 22 '17 at 13:52 ## 2 Answers I'll work in characteristic zero to avoid quirks due to the field. Suppose that V is an invariant subspace, and let v\in V. Assume for a moment that all entries of V are nonzero. We have that v,Tv,T^2v,T^3v,\ldots,T^{n-1}v\in V. If \alpha_0T^0v+\cdots+\alpha_{n-1}T^{n-1}v=0, we get a linear system with matrix$$ A=v_0v_1v_2\cdots v_{n-1}\begin{bmatrix} 1&1&1&\cdots&1\\1&2&3&\cdots&n-1\\ 1&2^2&3^2&\cdots&(n-1)^2\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ 1&2^{n-1}&3^{n-1}&\cdots&(n-1)^{n-1} \end{bmatrix} $$This is (one of) the well-known Vandermonde Matrix, which is invertible. So the only possible solution of the system is given by \alpha_0=\alpha_1=\cdots=\alpha_{n-1}=0, and thus v,Tv,\ldots,T^{n-1}v are linearly independent, which means that \dim V=n and so V=\mathbb F^n. So any invariant subspace will be made of vectors with at least one entry equal to zero. On the "nonzero part" of the subspace we can repeat the above reasoning, to conclude (as you suggested in your comment) that any invariant subspace of T is of the form$$ W=\text{span}\,\{e_j:\ j\in K\} $$for some K\subset\{1,\ldots,n\}. • :Thanks a lot... – JackT Oct 22 '17 at 14:07 Well, you can get 2^n (instead of just n) subspaces off the bat, by considering \operatorname{span}(B), where B ranges over all subsets of \lbrace e_1, \ldots, e_n \rbrace. Included in this are the whole space and the trivial space (spanned by the empty set), which are the range and nullspace respectively (unless the field has non-zero characteristic). The question is, are there any others? First, suppose \mathbb{F} = \mathbb{C}, and U is an invariant subspace, so that T|_U is an operator. It also will be diagonalisable. To see this, suppose \lambda is an eigenvalue for T|_U, and consider$$\operatorname{null}(T|_U - \lambda I|_U)^2 \subseteq \operatorname{null}(T - \lambda I)^2 = \operatorname{null}(T - \lambda I) = \operatorname{null}(T_U - \lambda I|_U),$$so the generalised eigenspace is no larger than the eigenspace. Hence a basis for$U$, consisting of eigenvectors for$T|_U$(and hence$T$) exists, so$U$must be of the above form. If$\mathbb{F} = \mathbb{R}$, the same argument mostly works. You just need to tiptoe around the complex case. If you extend the map to the complexification of the space, the argument works exactly as is. If$\mathbb{F}\$ is a different field, I'm not sure if I can help.

• You are absolutely correct. – JackT Oct 22 '17 at 14:11


# Is the set $\{2, 3, 4\}$ open in some metric spaces and not open in others?

I just want to check my understanding. This is from Baby Rudin:

2.18 Definition Let $$X$$ be a metric space. All points and sets mentioned below are understood to be elements and subsets of $$X$$.

$$(a)$$ A neighborhood of $$p$$ is a set $$N_r(p)$$ consisting of all $$q$$ such that $$d(p, q) for some $$r>0$$. The number $$r$$ is called the radius of $$N_r( p)$$

$$(e)$$ A point $$p$$ is an interior point of $$E$$ if there is a neighborhood $$N$$ of $$p$$ such that $$N \subset E$$

$$(f)$$ $$E$$ is open if every point of $$E$$ is an interior point of $$E$$.

Suppose we have the metric space with set $$X=\{1, 2, 3, 4, 5\}$$ and distance function $$d(x, y)=|x-y|$$. Now $$2$$ is an interior point of $$\{2, 3, 4\}$$ because $$N_{0.5}(2)=\{2\} \subset \{2, 3, 4\}$$ (and a similar argument can be made for $$3$$ and $$4$$ as well.

But if our metric space is $$\mathbb{R}$$ with the same distance function, then $$\{2, 3, 4\}$$ is not open because no neighborhood of $$2$$ is a subset of $$\{2, 3, 4\}$$, so $$2$$ is not an interior point of $$\{2, 3, 4\}$$, right?

• Right. That's completely correct. Sep 4 '17 at 16:01
• Why "no neighborhood of $2$ is a subset of $\{2,3,4\}$"? You are right, but you need to add some explanation. Sep 4 '17 at 16:01
• @Krish When we are dealing with the set $\mathbb{R}$, every neighborhood will contain numbers which are not integers.
– Ovi
Sep 4 '17 at 16:03
• @Krish: Because any neighborhood of $2$ (by the definition given) has the form $(2-r,2+r)$ for some $r>0.$ This will readily contain some non-integer rational number. Sep 4 '17 at 16:03
• @CameronBuie sorry!!! But I was just checking whether OP understood the reason clearly or not. (+1) for the question. Sep 4 '17 at 16:07

## 1 Answer

You're absolutely right. Nicely reasoned!

Another thing to consider is that even when the underlying space is the same, using a different metric may yield different open sets. Letting our metric space be $\Bbb R,$ but with the distance function $$\delta(x,y):=\begin{cases}0 & x=y\\1 & x\neq y,\end{cases}$$ we can show that (for example) $\{2\}$ is a neighborhood of $2$ with radius $\frac12,$ and by similar reasoning conclude that $\{2,3,4\}$ is once again open.


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Bohr Magneton Spin Magnetic Moment Of Lectron Calculator. iii) The bob of a conical pendulum under goes (A) Rectilinear motion …. 85 m and the angle with the vertical is 37°. A conical pendulum is a weight (or bob) fixed on the end of a string (or rod) suspended from a pivot. 2 The Conical Pendulum A small object of mass mis suspended from a string of length L. 10 L m a) Draw a labeled free-body diagram for the pendulum …. We will derive the equation of motion for the pendulum using the rotational analog of Newton's second law for motion about a fixed axis, which is …. Seat pan and sprinkle some the list …. In simple languages definition of a conical pendulum can be summed up as a conical pendulum consists of a weight (or bob) fixed on the end of a string or rod suspended from a pivot. Hang a bowling ball from a ceiling hook for a large conical pendulum to rotate at a steady slow speed. This motion of the pendulum is called motion of the first type. This video is made to understand Circular Motion. Rectilinear motion in vertical circle. Based on the previous notation, the following formulas can be obtained: Definition of average velocity. Standard conical pendulum lengths that have been investigated experimentally (and subsequently published) usually. A conical pendulum is a simple pendulum with the bob describing a circle and the string a cone · The centripetal force in a conical pendulum of semi vertical . It spins around the vertical with angle and with angular …. In the overhead view of above figure, a long uniform rod of mass 0. If during a periodic motion, the particle moves to and fro on the same path, the motion is vibratory or oscillatory. An elliptical (nonplanar) motion may precess due to anharmonicity in the restoring force. Basically, this is a mass on a string attached to a rubber stopper. The equation of SHM for a simple pendulum …. A conical pendulum has length 50 cm. A conical pendulum is a mass attached to a nearly massless string that is held at the opposite end and swung in . April 25, 2018 Boris Sapozhnikov. The actual form of a pendulum …. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum …. My Dashboard; Modules; WEEK 9 : CM2-CU8: Conical Pendulum; VIDEO : Conical Pendulum (Motion …. 2004): if the length of [a conical] pendulum be /, the semi-major axis of the ellipse described by the pendulum …. 12 S G Kamath The motion of a pendulum--be it the simple, spherical or the conical--is a concrete example wherein the three types of motions discussed above are realized physically. Because the bobs would lift in response to a faster speed (because they were basicly a conical pendulum…. What is a conical pendulum?. In the coordinate system that rotates with the wire, there will be fictitious Coriolis and centrifugal forces, in addition This Article is brought to you for free and …. This acceleration is called centripetal acceleration, and equals v 2 /r, where v is the speed of the object and r is the radius of. Suppose, further, that the object is given an initial horizontal velocity such that it executes a horizontal circular orbit of radius with angular velocity. The figure below shows a "conical pendulum", in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at …. THE COMPOUND PENDULUM The term “compound” is used to distinguish the present rigid-body pendulum from the “simple” pendulum of Section 3. Motion Of An Object Attached To A Spring. The plane of its motion, with respect to the earth, rotated slowly clockwise. Best answer Expression for tension in the string of a conical pendulum: i. To describe the motion of a conical pendulum in terms of its tangential velocity. Motion of a horse pulling a cart on a straight road. 68 m and the angle between the string and vertical is 35°. Conical pendulum illustrates uniform circular motion, and the other cases are representative of a non-uniform circular motion. A conical pendulum consists of a bob of mass 'm' revolving in a horizontal circle with constant speed 'v' at the end of a string of length 'l'. A) What is the magnitude of the torque (N. A simple pendulum consists of a relatively massive object - known as the pendulum bob - hung by a string from a fixed support. This result is true for all horizontal conical pendulums for which the angle, θ, is measured from the pendulum's position of vertical equilibrium. Please help me!!! I cnt solve thisA bob of mass is suspended from a fixed point with a masslessstring oflength (i. Rotational Inertia Stick Demo. The sections on mechanics in this book are basically arranged in that order. The pendulum motion is induced by the weight of the hanging mass, which is moved initially. let 'h' be the depth of the bob below the support. The deeper it sits, the wider the arch of the revolving pendulum, the slower the clock will go. Conical Pendulum: Its time period, tension i…. Cm 4 the conical pendulum (shared). Conical pendulum is a small heavy mass attached with massless, flexible and inextensible string suspended from a rigid support and the mass is constraint to whirl in a horizontal circle with constant speed. Fresnel Reflectance Of S Polarized Light. Oscillation motion is a periodic motion in which the particles move to and fro on a particular predetermined path within equal time intervals. In the table below, let us look at the various differences between simple and compound pendulum. I can go to a very small angle and then it's kind of barely moving, really slow like that. dimensions of the conical pendulum, the magnitudes of the forces acting during the conical pendulum motion and a triangular construction involving the conical pendulum period. Huygens rotated a conical pendulum of length l (when the angle θ was small) he. So a pendulum can evidently be said to have a certain amount of oscillatory motion in 1 direction plus a variable amount of angular momentum which results in a periodic amount of lateral motion in the orthogonal direction. A simple pendulum is an idealized body consisting of a particle suspended by a light inextensible cord. When pulled to one side of its equilibrium position and released, the pendulum swings in a vertical plane under the influence of gravity. Thus together with the string the bob traces out a cone. Example: Motion of moon around earth; Motion of a piston in a cylinder; Motion of a simple pendulum etc. What Is The Difference Between A Simple Pendulum And A Co…. that a conical pendulum whose initial motion was elliptical, was compelled to process in the same direction as the oscillation of its mass (Olsson 1978, 1981; Gray et arl. ERIC is an online library of education research and information, sponsored by the Institute of Education Sciences (IES) of the U. OCR past papers can be found at: https://www. Correct answer - The bob of a conical pendulum undergoes a) rectilinear motion in horizontal plane b) uniform motion in a horizontal circle c) The bob of a conical pendulum undergoes a) rectilinear motion in horizontal plane …. Location: Cabinet: Mechanic (ME) Bay: Shelf: #1 Abstract: A wire is suspended from a ceiling mount and a bowling ball attached at the bottom. (7) determines whether the plates of the pendulum …. Sneak a peak! Assault a graveyard. And silence those who arrive in the tuna. Derivation of the equations of motion. A motorized, plastic plane* is suspended from a thin string and “flies” in a circular path with a constant speed. The string is whirled in a horizontal circle, then the arrangement is called a conical pendulum. We first determine the time period of one complete rotation in conical pendulum. The characteristic of this motion can be obtained in terms of the length of the string and the angle with respect to the vertical. A particle of mass m, just completes the. Added an answer on August 11, 2020 at 1:32 pm. N2 - Students often find mechanics a difficult area to grasp. In this paper, we present evidence to show that the dynamics of rigid solid bodies is not a closed discipline, particularly in the field of …. ' Centripetal force (Fc) is the result of gravity and tension. The point of intersection satisfies the system of two linear equations: 12 12 57 22solutions …. Description : This set of 2 movies illustrates the dynamics of the conical pendulum (object suspended at the end of a string, moving in uniform circular motion). 22 A torsional pendulum consists of a rigid body suspended by a string or …. The time required for the bob of the conical pendulum to travel one revolution. Introduction The spherical pendulum is a mechanical system of considerable pedagogical interest [1–5]. Obtain an expression for its time period. Ballistic Pendulum lab report; Angular Motion lab report; Lab Report 9; Other related documents. 1 Use a search engine such as Google to research the history and uses of one of the following materials: Tin Glass Cement Titanium Carbon fiber Present the …. in the variable θ about the point (a near-conical pendulum). Determine the period of the pendulum …. In a conical pendulum, the bob is rotated with different angular v…. If the Lagrangian is approximated by keeping terms up to cubic order, the system has three independent constants of motion…. Torque and angular momentum of a conical pendulum joe_coolish May 19, 2011 May 19, 2011 #1 joe_coolish 9 0 Homework Statement A ball (mass m = 250 g) on the end of an ideal string is moving in circular motion as a conical pendulum as in the figure. Circular Motion A conical pendulum is a mass attached to a nearly mass-less string that is held at the opposite end and swung in horizontal circles. A motion of a Conical Pendulum. So, that's what I wanna talk to you about in this video. A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle, the motion of the particle …. 3: A torsional pendulum consists of a rigid body suspended by …. Circular MotionUnit: Circular MotionSubject: Physics Grade XI. Caption: A schematic diagram of "a conical pendulum in motion. In the corresponding motions of the pendulum, close to conical motion…. A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. Classify the following as motion along a straight line, circular or oscillatory motion: a. 75, and inverted pendulum, had a mass of 1000 kg and mechanical 1. [45] Researchers at the University of Chicago and Argonne National Laboratory have invented an innovative way for different types of quantum technology …. The conical pendulum was first studied by the English scientist Robert Hooke around 1660 as a model for the orbital motion of. The is an example of a 3D holonomic system. A special case of circular motion is the conical pendulum where the pendulum's "bob" is swung in a horizontal circle of radius Rwhile attached to a string of length L. If we take Tas time period of rotatory motion then velocity can also be expressed as: v= 2ˇr T: (2) Now replace the expression of vfrom Eq. A small ball of mass m is suspended from a string of length L. Draw a neat labelled diagram of conical pendulum. In this case, the moment of inertia has to be around the CM: I = 1 12 M L 2 leading to. Set the object traveling in a horizontal circular motion at a constant angle θ , as measured outward from when the object hangs straight down. A conical pendulum consists of an object moving in uniform circular motion at the end of a string of negligible mass (see Figure 1). Hang a bowling ball from a ceiling hook for a large conical pendulum …. Physics Lab 8: The Flying Pig – Centripetal Force Section: Name: Purpose To show the net force for a conical pendulum is mv 2 /r. uniform circular motion can be demonstrated with conical pendulum. Obtain the equations of motion of the system using Lagrange’s equations. I’m assuming that the pendulum bob is connected by an inelastic string of negligible mass, which allows me to fix the length of the string as. The correct option is : (a) Angular momentum of bob is …. You also should be able to find setup descriptions/figures in most textbook chapters on centripetal force. It is being regulated by a rotating pendulum that is called ‘conical’ after the shape of the silhouette of the motion. In this video, AAFREEN MAM has tried to explain the fundamentals necessary for solving physics questions. Of course, these different types of motion can be combined: for instance, the motion of a properly bowled bowling ball consists of a combination of trans-lational and rotational motion, whereas wave propagation is a combination of translational and oscillatory motion. The pendulum bob moves in a horizontal circle with a constant speed v. It is at ® rest under the action of …. Transcribed image text: PROJECTILE MOTION AND BALLISTIC PENDULUM Objective: The purpose of this experiment is to use the laws of conservation of energy and linear momentum to determine the velocity of a projectile, use this result to predict the projectile's range when fired in a uniform gravitational field and then compare this range to a measured value. 4b, which consisted of a particle at the end of a massless string. ANGULAR MOMENTUM OF CONICAL PENDULUM. Simple harmonic motion Solutions. Hello, The degree of freedom of a system = Number of directions in which it can move. When the string is horizontal there in …. In this Lesson, the sinusoidal nature of pendulum motion …. Introduction: A conical pendulum is a pendulum that is spun around in a circle instead of swinging backwards and forwards, such as like a traditional pendulum. A small mass is suspended by a cord and set into motion …. The pendulum does this because of inertia, which is the tendency of mass to stay in motion when a force acts upon it. For calculating the time period of a conical pendulum, we need to use the expression of Newton's second law of motion…. Answer (1 of 3): Consider a conical pendulum with a bob of mass m, length l, at an angle \theta with the vertical, going round with a uniform velocity v and …. Request PDF | The conical pendulum: The tethered aeroplane | The introductory physics lab curriculum usually has one experiment on uniform circular motion …. A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. The Conical Pendulum Download PDF A great activity for physics classes investigating centripetal force and uniform circular motion. Calculate… A simple pendulum mounted on a car. In this Lesson, the sinusoidal nature of pendulum motion is. To determine the acceleration due to gravity by means of a conical pendulum. Coupled pendula: Three varieties are available. February 15, 2018 Boris Sapozhnikov. Laboratory 8: Conical Pendulum –Activity L θ r Figure 2: Conical pendulum The conical pendulum, shown in Figure 2, con-sists of a mass m at the end of a …. 2:50 Breaking the force of tension into its components. Conical pendulum – measuring g Number 13573 0-EN Topic Me ch anics , two -dimensional motion Version 201 7-02-17 / HS Type Student exercise Suggested for grade 11 -12 p. Circular Motion and the Conical Pendulum Introduction. 5 THE CONICAL PENDULUM: A STUDY OF. A conical pendulum, a thin uniform rod of length. Consider about the figure of a conical pendulum. Kinetic energy = (1/2) Mω2(A2 – x2) Potential energy = 1/2 Mω2x2. The modulationequationsare found to be soluble in terms of elementary …. There's a 1:1 bevel that drives the pendulum. simple pendulum moves in to and fro motion. 0 degrees, then the angular speed of the conical pendulum …. Each of five modules contains reading links to a free textbook, complete video lectures, conceptual quizzes, and a set of homework problems. It doesn't take much effort to keep the mass moving at a constant angular velocity in a circle of constant radius. Take a video of the motion as viewed from the side. Ordinary pendulum clocks had been invented by Christiaan Huygens in 1656, and by ordinary, we mean that the pendulum swings back and forth in a vertical plane. With an elliptical path it represents a motion …. A heavy spherical mass (approximately 1. Estimation of parameters of various damping models in pla…. The problem of the conical pendulum is to consider a mass attached to one end of a light inextensible string of length with the other end attached at the top of a vertical rod. The path of periodic motion may be rectilinear, open/closed curvilinear. The motion of a simple pendulum is like simple harmonic motion …. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The three-dimensional motion of the elastic pendulum or swinging spring is investigated in this study. pendulum, body suspended from a fixed point so that it can swing back and forth under the influence of gravity. The swing of the pendulum will have to be limited to the distance of the support rods. motion answer key, describing and measuring motion, measuring motion gizmo lesson info explorelearning, download guided and study acceleration motion answers pdf, describing and measuring motion worksheet answer key 1, chapter 1 matter in motion section 1 measuring motion, describing motion …. The string makes a constant angle with respect to the vertical; as a result, the mass moves with constant speed in a horizontal circle. Equipment and Supplies Flying Pig (or equivalent), stopwatch, meter stick Discussion When an object travels at constant speed along a circular path, we say it has uniform circular motion …. In a typical experiment, the conical motion is confined to the horizontal plane, however, it is possible to extend the mathematical analysis to three dimensions (Barenboim & Oteo, 2013). The bob of a conical pendulum is attached to a fixed point A,A by a string of length 50, c, m,50cm and is moving in a circular path, as shown in the diagram . Uniform motion in a vertical circle. Discuss the pendulum's motion in the effective potential diagram. You can modify simulation parameters directly in MATLAB workspace. Physics Lab report on circular motion. Take your string back about 40 - 50 cm. The initial amplitude of the pendulum must be small compared to pendulum …. When the ball at the end of the string swings to its lowest point, the string is cut by a sharp …. For this exercise, assume that angles not small, and wº =2. This result is true for all horizontal conical pendulums for which the angle, θ, is measured from the pendulum…. Name: Laboratory Section: The goal of this laboratory is to study uniform circular motion. The pendulum is perhaps the simplest experimental devices ever constructed, and yet for all its simplicity it has historically enabled scientists to both investigate and enumerate gravity; the fundamental force that shapes the very universe. The study of this lab revolves around the generation, propagation and reception of mechanical waves and vibrations. This can be done side-by-side, front and back, clockwise, counterclockwise, in an elliptical motion, or even in a bobbing movement up and down, which often indicates a strong affirmative action. A simple pendulum performs simple harmonic motion about x = 0 with an amplitude a and time period T. When a pendulum swings in one direction and then comes back to its original starting point, this is called a period. Sir Isaac Newton (1642-1727) established the scientific laws that govern 99% or more of our everyday experiences. A conical pendulum is constructed by attaching a mass to a string 2. The time period of a simple pendulum: It is defined as the time taken by the pendulum …. The constraint on a bead on a uniformly rotating wire in a force free space is. 290 7 Lagrangian and Hamiltonian Mechanics 7. And I love these pure math ones. This work consisted of studying the motion of a rigid conical pendulum …. There are several different ways to configure a Twin Elliptic Pendulum…. ii) For a particle having a uniform circular motion, which of the following is constant (A) Speed (B) Acceleration (C) Velocity (D) Displacement. You have constructed what is known as a physical pendulum. Non-uniform circular motion Up: Circular motion Previous: Centripetal acceleration The conical pendulum Suppose that an …. 1k points) motion in a plane; class-11; 0 votes. We identified it from honorable source. When a pendulum is set in motion, gravity causes a restoring force that will accelerate it toward the center point, Conical Pendulum. Measure the period using the stopwatch or period timer. Slider-crank mechanism, arrangement of mechanical parts designed to convert straight-line motion to rotary motion, as in a reciprocating piston engine, or to convert rotary motion to straight-line motion, as in a reciprocating piston pump. ) Find an expression for v in terms of the geometry in Figure 6. Expression for its time period: Consider the vertical section of a conical pendulum having bob (point mass) of mass m and string of length ‘L’. Place the pendulum assembly on to hand driven rotator such that the set screw is tightened onto the flat part of the hand driven rotator. This resultant force is the centripetal force. The bob of a pendulum of length l is pulled aside from its equilibrium position through …. It is then put into an elliptical orbit thus acting as a conical pendulum. 500-m string is moving in a uniform circular motion in a horizontal plane of radius …. In 1851, Jean-Bernard-Leon Foucault suspended a 67 metre, 28 kilogram pendulum from the dome of the Pantheon in Paris. This necessary centripetal acceleration required for the motion of the conical pendulum is provided by the x component of tension which is the …. The motion of a pendulum is a classic example of mechanical energy conservation. An object at the end of a string is used as a conical pendulum. Course of Theoretical Mechanics, Higher Education Press, Bei Jing, Chapter 5. and the relevant moment of inertia is that about the point of suspension. The stiff string (actually a piece of aluminum wire) is required for otherwise the pendulum will not remain in the same (rotating) plane and the motion becomes closer to a spinning spherical pendulum. "A long-period conical pendulum for vibration isolation," Physics Letters A 222, 141-147 (1996). State technological advantages of Nano Moment of inertia, area meter 4m Moment of inertia, mass kilogram-meter 2kg m Momentum, linear kilogram …. 18 Determine the equations of motion …. Friction: Pulling a Box on a Horizontal Surface. The figure shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at a constant speed. Mass performing vertical circular motion under gravity. Do Problem 21 if the two masses are different. Science; Physics; Physics questions and answers; PROJECTILE MOTION AND BALLISTIC PENDULUM Objective: The purpose of this experiment is to use the laws of conservation of energy and linear momentum to determine the velocity of a projectile, use this result to predict the projectile's range when fired in a uniform gravitational field and then compare this range to a measured value. You must make a mark on the wall or your piece of paper to make sure that you let it go from the …. the motion of the mechanical system can be described by the generalized coordinates x ( t ) (translation M ), r ( t ) (elongation of the spring pendulum), and ˜ ( t ) (link rotation). The motion of conical pendulum is - 49146832 mukulsony1701 mukulsony1701 19. The string's motion follows a conical …. Vintage Jupiter Wall 31 Day Clock w/Pendulum and Key 26” Shop pendulum …. (You do NOT need to ﬁnd the frequency of the oscillations. Simulation- Conical Pendulum: 3D. The is horizontal, so there is no J verLieal motion…. (hint: first calculate the tension using the net y- forces = 0 and use newton's second law of motion …. Pendulums are also seen at many gatherings in eastern Mexico where they mark the turning of the tides on the day which the tides are at their highest point. The conical pendulum lab allows students to investigate the physics and mathematics of uniform circular motion. Static and Kinetic Friction on an Inclined Plane. He also explained our relationship to the Universe through his Laws of Motion …. A simple pendulum is a special case of a conical pendulum in which angle made by the string with vertical is zero i. 130 m) for the conical pendulum…. 6-53 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. The pendulum is a simple mechanical system that follows a differential equation. A physical pendulum consists of a uniform rod of length d and mass m pivoted at one end. Additionally, what are the 4 types of motion? These four are rotary, oscillating, linear and reciprocating. pendulum mass for a fixed length, and varied the pendulum length for a fixed mass. The simple pendulum is another mechanical system that moves in an oscillatory motion. g = acceleration due to gravity. This means that the forces in line with the arm of the pendulum must be equal and opposite since there is no motion in this direction and we see that T = m g cos. The goals of this lab are to verify that centripetal acceleration is given by a = v 2 /r and to show that the period of a conical pendulum is given by the theoretical equation:. There are two forces acting on the bob: its weight and the tension in the string (Figure 39). Determine the horizontal and vertical components of the force exerted by the wire on the pendulum…. oPhysics: Interactive Physics Simulations. An analysis of the motion presents that the equilibrium states of the pendulum are determined by the pendulum angular speed. Other rides, such as the rotor ride, Enterprise wheel, and Ferris wheel, spin the rider in circular motion either horizontally or vertically. The pendulum consists of a string and a bob (a weight, generally spherical) at the end. The reasons for this, however, and consequently the focus …. Robert Hooke's conical pendulum from the modern viewpoint of. A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown in the figure. Let m be the lower mass and let M be the sum of the two masses. The motion of a conical pendulum in a rotating frame. What tangential speed, v, must the bob have so that it moves in a horizontal; Question: To describe the motion of a conical pendulum in terms of its tangential velocity. In both cases, an elliptical motion is induced. Find the maximum height above the ground that the ball reaches. Instead of swinging back and forth, the bob is to move in a horizontal circle at constant speed v, with the wire making a fixed angle β with the vertical direction (Fig. It is Robert Hooke who first studied the motion of a spherical pendulum in that a conical pendulum whose initial motion was elliptical, . Theoretically, The time period of a conical pendulum is directly proportional to the square root of its length. pendulum altogether in absolute measurements of g. The oscillatory motion of a simple pendulum: Oscillatory motion is defined as the to and fro motion of the pendulum in a periodic fashion and the centre point of oscillation known as equilibrium position. State the expression for its periodic time in terms of length. The mass is moving about the rod in uniform circular motion …. In your text, it is shown that the period of rotation T of a conical pendulum of length, L, is given by € T=2π Lcos(θ) g, where θ is the angle the pendulum …. A simple pendulum consists of a point mass suspended on a string or wire that has negligible mass. As the Bravais pendulum is a conical pendulum oscillating at Earth's surface, its motion is also discussed. Its construction is similar to an ordinary . 6-53 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal …. The conical pendulum was first. ω: Angular velocity of rotational motion (rad/s) T: period of rotational motion (seconds) Find the period of a conical pendulum. The string makes an angle θ with the vertical. All of the energy in the pendulum is gravitational potential energy and there is no kinetic energy. The pendulum always moves in one angular direction. L2 : Motion of Pendulum - Motion and Time, Science, Class 7; Video | 04:40 min. Popular; Recent; Comments; Pendulum with spring animation MATLAB. The motion occurs in a vertical plane and is driven by a gravitational force. The moment of intertia I for a point mass rotating around a pivot (radius l) is ml², yielding our equation of motion: This is a second-order, non …. ࡱ > 3 5 2 #` p bjbjm m 5 6 > > > x , Q D D D D H S U U U E $h o ~ E> D H ; 7 7 7 @ 8D > H S 7 S 7 7 & > 7 H 8 @h ?Z 7 S ! 0Q 7 p 7 > 7 7 7 Q$ R f t AP Physics Lab Brockport High School NY USA Circular Motion: The Conical Pendulum Mr Keefer Objectives: Determine the acceleration of gravity g from the circular motion of a conical pendulum…. swing with horizontal circular motion such that r equals the length of the pendulum?. This pendulum is used as a model to analyze the motion of planets. Korean pendulum wall clocks. The reason that the pendulum oscillates about the vertical is that if the pendulum is displaced, the force of gravity pulls down on the pendulum. Expression for Period of Conical Pendulum: Let us consider a conical pendulum consists of a bob of mass 'm' revolving in a horizontal circle …. The time it takes the pendulum …. The period of the motion of a pendulum is virtually independent of its amplitude and depends primarily on the geometry of the pendulum …. Question 3 The figure shows a conical pendulum, in which th…. Consider a small body of mass m suspended from a rigid support with the help of a string of length l. Consider a conical pendulum with a line length ‘L’ and a rotation radius ‘r. Conical Pendulum Time Period: It consists of a string OA whose upper-end O is fixed and bob is tied at the other free end. The forces which are acting on the mass. The period of a simple pendulum …. The ball revolves with constant speed v in a horizontal circle of radius r as shown in the figure. In this experiment a mass is attached to a string and made to spin in a circle of fixed radius, the time period of the motion …. Correction of the excercise 4:00. Lab – Flying Pigs Purpose: To show that the net force for a conical pendulum is mv 2 /r Equipment/Supplies: flying pig apparatus stopwatch meterstick Discussion: Any object moving in uniform circular motion …. Equations for the following principal physical parameters. The restoring torque is supplied by the shearing of the string or wire. In other words, the vertical component of the tension force . The bob does not show any vertical motion …. The length L of the string is 1. A Bob of a conical pendulum undergoes option A rectilinear motion in a horizontal plane option B uniform motio… Get the answers you …. Differential equation of the motion (derived from Newton's second law):. This text is useful for students to follow many details on the motion of the Foucault pendulum in inertial and rotational frames and help in the study of the particle motion …. • As viewed from above, it moves in a circle, the centripetal force. The dimensionless factor of $$2\pi$$ can be derived using an in-sight from Huygens [15, p. doing this isaac physics question on circular motion, can anyone help The bob of a conical pendulum is attached to a fixed point A,A by a string of length. The pendulum is initially at rest in a vertical position. Imagine a conical pendulum in steady circular motion with small angle θ. The bob moves in a single plane, back and forth. This can be determined by measuring the time required for the pendulum to reoccupy a given position. (a) Write a MATLAB script, which solves (*) for. The equations of motion of the pendulum were derived using the Lagrangian method. The motion of the pendulum depends on its total energy E = T +V. 9 0 m and negligible mass and the bob follows a circular path of the circumference. eu Viaduktvej 35 · DK-6870 Ølgod Fax +45 7524 6282 www. The canonical example of simple harmonic motion is the motion …. This is a simulation of a double pendulum. Spherical pendulum and vertical pendulum are the special cases of conical pendulum. Geometrically, the arc length, s, is directly proportional to the magnitude of the central angle, θ, according to the formula s = rθ. October 28, 2018 Boris Sapozhnikov. As a result, the motion of the pendulum …. n Review Part A (Figure 1)A massless string of length L is fixed to a point in the ceiling and suspends a bob with mass m (i. h is the distance from the plane of the circular motion to where the string is attached. Is there any limitation on semivertical angle in a conical pendulum? Thank You. study of circular motion in conical pendulum purpose the purpose of this experiment is to study the effect of the. Class 7 Science Chapter 13 Motion and Time Textbook Exercise Questions and Answers. The 1 ∶ 1 ∶ 2 resonant elastic pendulum is a simple classical system that displays the phenomenon known as Hamiltonian monodromy. Let the bob be displaced from its mean position and whirled around a horizontal circle of radius 'r' with constant angular velocity 'ω'. If the weight is made to describe a horizontal circle so that the string describes a right circular cone, whose axis is a vertical through ( O ) , then system is called a conical pendulum. A new twist for the conical pendulum A new twist for the conical pendulum Moses, Thomas; Adolphi, Natalie L. Then the period of the simple pendulum is given by. We use a conical pendulum in this experiment. Jean Bernard Léon Foucault was the leading experimental physicist of his day. Imagine a basic conical pendulum. In this work, planar free vibrations of a single physical pendulum are investigated both experimentally and numerically. But, because for small θ, we can say that θ ≈ sinθ, we can re-write the DE as:. You can apply similar considerations to a simple pendulum, which is one on which all the mass is centered on the end of a string. The equations for a simple pendulum show how to find the frequency and period of the motion. This allowed the reduction of apparent gravity, thus increasing its natural period, a similar approach to that of Galileo more than two centuries earlier when he used an inclined plane to study the motion of falling objects. Equations for a Simple Pendulum. An inverted pendulum is simply a pendulum which has its fixed end located below the vibrating mass. Let us consider a conical pendulum …. The pendulum is in steady circular motion with constant angular velocity wk. the pendulum bob has an initial velocity. The Chaotic Motion of a Double Pendulum. In two or three attempts one could realize a satisfactory circular orbit. Other rides, such as the rotor ride, Enterprise wheel, and Ferris wheel, spin the rider in circular motion …. When this happens the pendulum is called a "conical pendulum" because the cord sweeps out (or creates in space) a conical surface with the apex at the …. Two pendula coupled by a spring (shown to the right) will show normal modes, and transfer of energy between the single pendula swinging modes. One end of a string ( AO ) is attached to a fixed point ( O ) and a weight ( W ) is tied to the other end ( A ). Uniform Accelerated Motion. Use the Tracker Software to analyze the motion of a conical pendulum, as viewed from the side. Let – ( N ) is the centre of horizontal circle of radius ( r ) …. Exploring Projectile Motion Concepts. Let the string subtends an angle θ with the vertical. We use centripetal acceleration here, because we have a circular motion. Conical pendulum clocks are part of the clock history but as they are not easy to implement you don’t see too many around, and as far as I know there are no wooden conical pendulum clocks. In this lab you will • test the theory of simple harmonic motion in the case of a simple pendulum and a Hooke’s law spring. The kinetic energy would be KE= ½mv2 ,where m is the mass of the pendulum, and v is the speed of the pendulum. Consider an object of mass m tied to a string of length l and whirled in a horizontal circle of. Because the bobs would lift in response to a faster speed (because they were basicly a conical pendulum) it could be. The upward force produced by the tension balances the downward force from the weight of the pendulum:. 2 is annotated with the mathematical symbols that will be used in the analysis of the period of the pendulum. The massless thread is only an idealization. A simple pendulum theoretically has the mass of the bob concentrated at one point, but this is impossible to achieve exactly in practice. 17 A simple pendulum of length l and mass m is pivoted to the block of mass M which slides on a smooth horizontal plane, Fig. Physics Lab 9: The Flying Pig – Centripetal Force Section: Name: Purpose To show the net force for a conical pendulum is mv 2 /r. Once the modules are completed, the course ends with. A ball is attached to a string and swung so that it travels in a horizontal circle. pendulum synonyms, Compensation pendulum; compound pendulum; Conical pendulum; and twisting his legs round it in sailor fashion, slipped down eight or ten feet, where his weight gave it a motion not un-like that of a pendulum…. tal plane with its string sweeping the. Conical Pendulum – If a simple pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called a conical pendulum. The accel-eration is directed along the radius of the dashed circle and towards the shaft. He realized that the pendulum …. Periodic Motion Lab – The Conical Pendulum Purpose The goals of this lab are to verify that centripetal acceleration is given by a = v2/r and to show that the period of a conical pendulum is given by the theoretical equation: Procedure A small mass is suspended by a cord and set into motion …. Experiment with Conical Pendulum. In the figure below, a simple pendulum is represented at various positions of its motion. We have two contributions: kinetic energy of rotation around the CM and kinetic energy …. Unfortunately, Java cannot plot the motion of the pendulum just by using the angle q – it uses (x, y) coordinates to plot shapes. VIDEO : Conical Pendulum (Motion in a Horizontal Circle) Skip To Content. It is the analogue of the conical motion of a spherical pendulum with a fixed point of suspension. Smart pendulum leveling system One-button operation 530 Ft. motion of the conical pendulum is just the sum of the motion of two pendulums. In conical pendulum the bob does not oscillate back and forth but it moves in a circle. A conical pendulum is a mass attached to a nearly massless string that is held at the opposite end and swung in horizontal circles. Find Physics textbook solutions? 500 Selected Problems In Physics for JE… 877 solutions Selina - Concise Physics - Class 9 1224 solutions Lakhmir Singh, Manjit Kaur - Physics 10.


# Proving units in a ring

Suppose $R$ is a ring with no zero divisors and with identity $1_R$ not equal to $0_R$. Suppose that $a,b$ are in $R$ and that $ab$ is a unit. Prove that $b$ is a unit.

My thoughts: I know a unit is basically a unit that (for this example) would mean $abu = 1_R$ for some nonzero $u$ in $R$. I am really stuck after that. Not seeing a clear path to manipulate the variables to prove b is a unit by itself.

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Well, write it this way: $\;1=u(ab)=(ua)b\implies b\;$ is a unit...:) –  DonAntonio Apr 27 at 19:00
@Don That yields only a left inverse $\,c = ua\,$ for $\,b.\,$ But conjugation shows it is a right inverse too - see my answer. –  Bill Dubuque Apr 27 at 23:25

As the others have pointed out the calculation $$1=u(ab)=(ua)b$$ shows that $ua$ is a left inverse to $b$. Consider the product $b(ua)$. We have $$ua=1(ua)=((ua)b)(ua)=(ua)(b(ua)),$$ so $$(ua)(1-b(ua))=0.$$ As the ring has no zero divisors this implies that either $ua=0$ or $b(ua)=1$. But if $(ua)=0$, then $1=(ua)b=0$ which is a contradiction. The claim follows.

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+1 for actually using the fact the ring has no zero divisors... –  Goos Apr 27 at 19:30
@Goos: Well, the claim is false without something extra, and that was the only piece available :-) All: Sorry about spoiling the problem. I really should have come up with an appropriate hint. –  Jyrki Lahtonen Apr 27 at 19:45
yes, but there were two maybe three people claiming "solutions" without using that fact. So +1 for a legitimate proof that it is a unit and not just having a left inverse. –  Goos Apr 27 at 20:10
@Jyrki Re: hints. See my answer for one way I often hint at it. It's not easy to give a hint for this without spilling the beans. –  Bill Dubuque Apr 27 at 23:05

As discussed in the comments, since $ab$ is a unit then $uab = 1$ for some $u \in R$, so $ua$ is a left inverse for $b$. It remains to show that $ua$ is also a right inverse for $b$, i.e., $bua = 1$. Taking the equation $1 = uab$ and multiplying both sides by $ua$ on the right, we have $$ua = uabua \implies 0 = ua - uabua = ua(1 - bua) \, .$$ Since $R$ has no zero divisors, then either $ua = 0$ or $1 - bua = 0$. But again, $R$ has no zero divisors, so we must have $1 - bua = 0$, hence $1 = bua$. Thus $ua$ is a two-sided inverse for $b$.

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Hint $\$ Conjugate a one-sided inverse $\,bc=1\,$ to the other side via $\ (bc\!-\!1)b\, =\, b(cb\!-\!1)$

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# Thread: centre of a group Z(G)

1. ## centre of a group Z(G)

Hey there, i have a question on the center of a group.

The centre Z(G) of a group G is defined by $\displaystyle Z(G) = g \epsilon G: \forall x \epsilon G, xg = gx$

(i) Show that Z(G) is normal subgroup of G
(ii) By considering the Class Equation of G acting on itself by conjugation show that if $\displaystyle |G| = p^n$ ( p prime) then $\displaystyle Z(G) \neq {1}$

(iii) If G is non abelian show that G/Z(G) is not cyclic.
(iv) Decude that any group of order $\displaystyle p^2$ is abelian.
(V) Deduce that a gorup of oder $\displaystyle p^2$ is isomorhpic either to $\displaystyle C_{p2}$ or to $\displaystyle C_p \times C_p$

any hints would be greatly appreicated and ill atempt the question once i have a clear idea of how to do them. consequently, i will post them online once i have worked them out. cheers guyz

2. Originally Posted by joanne_q
(i) Show that Z(G) is normal subgroup of G
Let $\displaystyle x\in \text{Z}(G)$ prove that $\displaystyle gxg^{-1} \in \text{Z}(G)$ for any $\displaystyle g\in G$.

(ii) By considering the Class Equation of G acting on itself by conjugation show that if $\displaystyle |G| = p^n$ ( p prime) then $\displaystyle Z(G) \neq {1}$
The conjugacy class equation says,
$\displaystyle |G| = |\text{Z}(G)| + \sum [G:\text{C}(x)]$.
Where the $\displaystyle x$'s are taken from distinct conjugacy classes of more than one element.
We know that the left hand side is divisible by $\displaystyle p$ so the right hand side is divisible by $\displaystyle p$. Now $\displaystyle \text{C}(x)$, the centralizer, is not whole $\displaystyle G$ because we are picking those $\displaystyle x$, this means $\displaystyle 1\leq |\text{C}(x)| \leq p^{n-1}$. This means $\displaystyle p$ divides the index of $\displaystyle \text{C}(x)$ under $\displaystyle G$. Thus, this proves that $\displaystyle \text{Z}(G)$ is divisible by $\displaystyle p$. Thus, the center is non-trivial.

3. Originally Posted by joanne_q
(iii) If G is non abelian show that G/Z(G) is not cyclic.
Contrapositive is easier. Let $\displaystyle H=\text{Z}(G)$. If $\displaystyle G/H$ is cyclic then there is $\displaystyle aH$ which generates the group $\displaystyle G/H$. Let $\displaystyle x,y\in G$. Note $\displaystyle xH,yH\in G/H$ thus $\displaystyle xH=a^nH$ and $\displaystyle yH=a^mH$. This means $\displaystyle x = a^n z_1$ and $\displaystyle y=a^mz_2$ where $\displaystyle z_1,z_2\in H$. But then $\displaystyle xy = a^n z_1 a^mz_2 = a^{n+m}z_1z_2$ and $\displaystyle yx = a^m z_2 a^n z_1 = a^{n+m}z_1z_2$ because $\displaystyle z_1,z_2$ commute with everything. Thus $\displaystyle G$ is abelian.

(iv) Decude that any group of order $\displaystyle p^2$ is abelian.
By Burnside's lemma (that is (ii)) we have that the center is non-trivial, forming the factor group we have a cyclic group. Thus the original needs to be abelian.

4. Originally Posted by joanne_q
(V) Deduce that a gorup of oder $\displaystyle p^2$ is isomorhpic either to $\displaystyle C_{p2}$ or to $\displaystyle C_p \times C_p$
Let $\displaystyle |G|=p^2$. Pick $\displaystyle a\not = 1$. Form the subgroup $\displaystyle H=\left< a \right>$ if $\displaystyle H = G$ then the group is cyclic and proof is complete. Otherwise choose $\displaystyle b\in G\setminus H$ and form $\displaystyle K=\left< b\right>$. Now $\displaystyle H\cap K = \{ 1\}$ this means $\displaystyle HK = G$*. Also $\displaystyle H,K\triangleleft G$ because the group is abelian. This means $\displaystyle G\simeq H\times K \simeq \mathbb{Z}_p \times \mathbb{Z}_p$.**

*)Because $\displaystyle |HK| = |G|$ by using the fact $\displaystyle |HK||H\cap K| = |H||K|$.

**)Theorem: If $\displaystyle H,K$ are normal subgroups with $\displaystyle H\cap K = \{ 1 \}$ and $\displaystyle HK = G$ then $\displaystyle G\simeq H\times K$.

5. Thanks a lot for the help really appreciate it. makes it easier to understand the subject aswell.

For part (ii), your solution was:
Originally Posted by ThePerfectHacker
The conjugacy class equation says,
$\displaystyle |G| = |\text{Z}(G)| + \sum [G:\text{C}(x)]$.
Where the $\displaystyle x$'s are taken from distinct conjugacy classes of more than one element.
We know that the left hand side is divisible by $\displaystyle p$ so the right hand side is divisible by $\displaystyle p$. Now $\displaystyle \text{C}(x)$, the centralizer, is not whole $\displaystyle G$ because we are picking those $\displaystyle x$, this means $\displaystyle 1\leq |\text{C}(x)| \leq p^{n-1}$. This means $\displaystyle p$ divides the index of $\displaystyle \text{C}(x)$ under $\displaystyle G$. Thus, this proves that $\displaystyle \text{Z}(G)$ is divisible by $\displaystyle p$. Thus, the center is non-trivial.

here is my solution, is it correct aswell?

$\displaystyle G \equiv |Z(G)| (mod p)$ since Z(G) is a fixed point set.
Now $\displaystyle |Z(G)| \equiv p^n(mod p)$, |Z(G)|=0.
So Z(G) has atleast p elements.

6. for part (iii) i believe you have proved the opposite of the question? i.e. G/Z(G) is cyclic and thus abelian...

to prove that it is non cyclic, do all the points you stated have to be contradicted..?
Originally Posted by ThePerfectHacker
Contrapositive is easier. Let $\displaystyle H=\text{Z}(G)$. If $\displaystyle G/H$ is cyclic then there is $\displaystyle aH$ which generates the group $\displaystyle G/H$. Let $\displaystyle x,y\in G$. Note $\displaystyle xH,yH\in G/H$ thus $\displaystyle xH=a^nH$ and $\displaystyle yH=a^mH$. This means $\displaystyle x = a^n z_1$ and $\displaystyle y=a^mz_2$ where $\displaystyle z_1,z_2\in H$. But then $\displaystyle xy = a^n z_1 a^mz_2 = a^{n+m}z_1z_2$ and $\displaystyle yx = a^m z_2 a^n z_1 = a^{n+m}z_1z_2$ because $\displaystyle z_1,z_2$ commute with everything. Thus $\displaystyle G$ is abelian.

7. Originally Posted by joanne_q
For part (ii), your solution was
Have you ever done the conjugacy class equation? There is a way around it if you never done it that way, I think I have an idea of what you might have done.

Let $\displaystyle G$ be a finite $\displaystyle p$-group. Let $\displaystyle G$ act on a non-empty finite set $\displaystyle X$. Then $\displaystyle |X|\equiv |X^G|(\bmod p)$ where $\displaystyle X^G$ is the invariant subset fixed by $\displaystyle G$.

here is my solution, is it correct aswell?

$\displaystyle G \equiv |Z(G)| (mod p)$ since Z(G) is a fixed point set.
Now $\displaystyle |Z(G)| \equiv p^n(mod p)$, |Z(G)|=0.
So Z(G) has atleast p elements.
Let $\displaystyle G$ act on itself by conjugation (i.e. $\displaystyle X=G$ and $\displaystyle g*x = gxg^{-1}$). Also $\displaystyle G$ is a finite $\displaystyle p$-group which fits the above result thus $\displaystyle |G| \equiv |G^G| (\bmod p)$ but $\displaystyle G^G = \text{Z}(G)$ because that is the subset left fixed under conjugation. Thus, $\displaystyle |G| \equiv |\text{Z}(G)|(\bmod p)$ which means the center needs to be divisible by $\displaystyle p$, i.e. it cannot be trivial.

8. Originally Posted by joanne_q
for part (iii) i believe you have proved the opposite of the question? i.e. G/Z(G) is cyclic and thus abelian...

to prove that it is non cyclic, do all the points you stated have to be contradicted..?
No, there is no contradiction argument. I proved the contrapositive statement.