Problem 5.25 If B is uniform, show that A(r) = - (r x B) works. That is, check that V · A = 0 and V × A = B. Is this result unique, or are there other functions with the same divergence and curl?

For Problem 8.48, am I to find these two things via taking the derivatives?

For Problem 8.16, how do I prove the relations and give the correct expressions?

1 W:0E *Problem 1.3 Consider the gaussian distribution p(x) = Ae¬^(x-a)² %3D where A, a, and A are positive real constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. (b) Find (x), (x²), and ơ. (c) Sketch the graph of p(x).

A pendulum of length l and mass m is mounted on a block of mass M. The block can movefreely without friction on a horizontal surface as shown in Fig 1 PHYS 24021. Consider a particle of mass m moving in a plane under the attractive force μm/r2 directedto the origin of polar coordinates r, θ. Determine the equations of motion.2. Write down the Lagrangian for a simple pendulum constrained to move in a single verticalplane. Find from it the equation of motion and show that for small displacements fromequilibrium the pendulum performs simple harmonic motion.3. A pendulum of length l and mass m is mounted on a block of mass M. The block can movefreely without friction on a horizontal surface as shown in Fig 1.Figure 1Show that the Lagrangian for the system isL =( M + m2)( ̇x)2 + ml ̇x ̇θ + m2 l2( ̇θ)2 + mgl(1 − θ22)

Three identical cylinders of radius r are placed inside a hollow cylinder of radius R. All cylinder axes (perpendicular to the paper) are horizontal. There is no friction. The cylinders B and C are on the verge of separating (= infinitesimally separated, as shown). A B (a) From the statics equations for A and B, show that the angle between the normal under B and the vertical is given by tan 0 1 (The same result is obtained for A and C, of course, since B 3/3 and C have mirror image forces on them.) (b) By trigonometry of geometry, show that R must be r(1+2/7 Jin order for B and C to be on the verge of separating. (Find sin 0 and cos 0 from a triangle; don't find 0.) (Problem from a senior-year high school physics book used in England.) ogbo

(a) Discuss your understanding of the concepts of the symmetry of a mechanical system, a conserved quantity or quantities within the mechanical system and the relation between them. Illustrate your answer with an example, but not the example in the Lecture Notes. What is the benefit of symmetry when analysing a mechanical system? (b) Consider the Lagrangian function on R? (defined by the Cartesian coordinates (x, y)) given by 1 L m (i² – ý²) + a(y² – x²), where m and a are constants. (i) Show to first order in e (that is, ignore terms of order e? and higher), that L is invariant under the transform (x, y) + (x + €Y, Y + ex). (ii) Find the integral of motion predicted by Noether's theorem for the Lagrangian function L.

(a) Let F₁ = x² 2 and F₂ = x x + y ŷ + z 2. Calculate the divergence and curl of F₁ and F₂. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. (b) Show that the field F3 = yz î + zx ŷ + xy 2 can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface.   Here, the total force acting on the mass which is on the table is -T r^ (r hat). Why?

Problem 4.16 It is desired to find the equation for the shortest distance be- tween two points on a sphere. Determine the functional for this problem. (Use spherical coordinates.)