PHAS0010 - Classical Mechanics

1. (a) A uniform disk of mass M and radius R is attached to a wall with a spring of constant k. The disk is pulled a distance A from the spring’s equilibrium position and then released as shown below. The disk is rolling without slipping.

i. What is the maximum speed of the disk? [3]

ii. What is the angular frequency ω of the harmonic motion? [1]

iii. What would the spring constant be for this system to have the same ω as a system comprised of a frictionless non-rolling block of mass M and spring with stiffness k? [1]

(b) A hole of radius r has been drilled in a uniform disk of radius R. The centre of the hole is at a distance d from the centre of the disk as shown in the figure below. The mass of the disk without the hole is M. Find the moment of inertia for rotation about an axis perpendicular to the plane of the disk and passing through its centre. [5]

(c) A horizontal disk is rotating with a constant angular speed ω about its axis of symmetry as shown in the figure below. A point mass m is launched directly from the centre of the rotating disk with the initial velocity v0. The motion of the mass is frictionless.

i. Consider the frame associated with the rotating disk and write down all the fictitious forces acting on the moving mass m using the polar coordinate system at the position ~r = (r, θ) with clear indication of their directions. [3]

ii. Show that the polar coordinates r and θ describing the instantaneous position of the mass on the rotating disk obey the following equations of motion: [3]

iii. Assuming ¨θ(t) = 0 solve these equations for r = r(t) and θ = θ(t) using the initial conditions stated above. What is the trajectory of the point mass as it is moving across the disk? [4]

2. (a) Two identical objects of mass m are separated by a distance d and interact through the gravitational force only. The objects are given initial velocities v0 as shown in the figure.

i. Obtain a condition for the objects to follow circular orbits of diameter d, expressed in terms of G, m, d and v0. [3]

ii. Write down the total energy of the system and hence obtain a condition for objects to follow closed orbits, in terms of G, m, d and v0. [3]

(b) Derive the moment of inertia of a uniform rod of mass m and length l about an axis which passes through one end of the rod and makes a constant angle α with the rod as shown in the figure. [4]

(c) A cable drum on which a cable is wound consists of a central hollow thinwalled cylinder of mass M and radius R and two solid circular uniform end plates, each of mass M and radius 2R, as in the diagram below. The cable drum, with several turns of thin massless cable wrapped round it is placed on a horizontal surface. The end of the cable feeding off the bottom of the drum is connected to a block of mass m though a massless and frictionless pulley. The drum rolls without skidding with acceleration aD.

i. Determine the moment of inertia of the cable drum about AB, the axis through the centre. Hint: see exam rubric for the moments of inertia of different objects. [1]

ii. Sketch the free-body diagram and indicate the forces acting on the cable drum. [1]

iii. State the equations of motion for [2]

• the rectilinear motion of the centre of mass of the drum;

• the rotational motion of the drum about the axis AB;

• the vertical motion of the block m.

iv. Hence determine, in terms of m, M and g [6]

• the force of friction between the drum and the ground;

• the force of tension from the cable on the drum;

• the linear acceleration aB of the block m;

• the linear acceleration aD of the center of mass of the drum. In which direction the drum moves?

3. (a) A block explodes into three parts of equal masses m as shown in the diagram, with the mass travelling to the right going at speed v.

i. Derive the relations between the speeds of the three masses after the explosion. [2]

ii. Hence obtain the energy released by the explosion in terms of m and v. [2]

(b) Two small blocks, both of mass m, are initially at rest on the two sides of the larger triangular block of mass M as shown in the figure. The smaller blocks start sliding down the frictionless sides of the larger block. The angles of the triangle block are α, β and 90◦ , as shown below.

i. Find the normal forces of each small block on the larger block. [1]

ii. Find the force that the table-top exerts on the larger block M in its simplest form. [3]

iii. Find the acceleration of the larger block relative to the table top considering that all contact surfaces are frictionless. [2]

(c) A small object of mass m can slide without friction inside a smooth cylindrical well of radius r. It is placed on the top of the well such that θ = 0 and begins to slide down one side of the well under the influence of the gravitational acceleration g.

i. State the forces acting on the mass giving their directions as it slides on the well. Write down the radial and angular (transverse) components of the equation of motion in polar coordinates. [4]

ii. With the help of the Work-Energy theorem or otherwise, show that

Hence obtain the reaction force between the mass and the surface of the well as a function of θ. [4]

iii. What can you say about the conservation of the angular momentum of the object? [2]