Advanced General Physics Homework Assignment 4, Fall 2023

Advanced General Physics
Homework Assignment 4, Fall 2023
Due on Wednesday, October 11 at 11:59 am

Problem 1

Consider a pendulum made of a spring with a mass m on its end. See the figure. The equilibrium length of the spring is l. Let the length of the spring be l + x and its angle with the vertical be θ. Assuming that the motion takes place in a vertical plane, use the Euler-Lagrange equations to find the equation of motions for x and θ. That is find expression for ¨x and θ ¨.

Problem 2

Spherical coordinates are r, θ and φ where r is the distance from the center, θ is the angle with respect to the North pole (zenith angle) and φ is the angle around the equator (azimuthal angle). See the figure below. 

The kinetic energy of a particle in spherical coordinates is given by:


Consider a particle of mass m moving in a potential energy that depends only upon r and θ, U(r, θ)

a) Find the expression for the Lagrangian.

b) Find expression for the momenta conjugate to r, θ and φ.

c) Which of the conjugate momenta are conserved?

d) Using the Euler-Lagrange equations find the equations of motion. That is, find expressions for ¨ r, θ ¨ and ¨ φ.

Problem 3

A ball of mass m is whirled around on a string which passes through a ring as shown in the figure. Neglect gravity. At t = 0, the ball is rotating at an angular velocity˙θ = ω0. The string is pulled with constant velocity ˙r starting at t = 0 so that the radial distance of the ball from the center decreases. The general equations of motion
for r and θ that result from the Euler-Lagrange equations are:

Use these to find:

a) The force on the string at time t = 0

b) The rate at which the force is increasing at time t = 0

Problem 4

A block of mass m is held motionless on a frictionless plane of mass M and angle of inclination θ as shown in the figure below. The plane rests on a horizontal frictionless surface. The block of mass m is then released. Let x1 be the horizontal position of the plane and let x2 be the horizontal position of the block relative to the plane. Also note that there is a relation between the vertical position of the of the block, y2 and its horizontal position with respect to the plane, x2.

2023-10-14