ASSIGNMENT-III

PH 201 CLASSICAL MECHANICSFULL MARKS: 85

GIVEN DATE: 15/10/14DUE DATE: 29/10/14

(1) Show that a particle moving in one dimension subject to the potential V (x) = k|x|p,where k > 0 and p > 1, will oscillate with a period proportional to a−(p−2)/2,where a is the amplitude. (The particle oscillates back and forth between the pointsx = −a and x = a with a total energy E = kap). 4

(2) Consider a small spherical ball (of uniform mass density) rolling back and forthdue to gravity near the bottom of a circular track

whose radius is R as shown. Assume that

the ball rolls without slipping. Calculate

the time period of small oscillations. 4

(3) A particle of mass m is constrained to move on a curve in the vertical plane definedby the parametric equations

x = l (φ + sinφ) and z = l (1 − cosφ) ,

where −π < φ < π. If the particle is subject to the force of gravity, calculate thetime period of oscillations as a function of the amplitude φ0 (assumed to be lessthan π).

Note: This is the equation of a cycloid which is the curve traced out by a pointfixed to the circumference of a circle of radius l rolling along the x-axis. Thereis no restriction on φ for the rolling motion, and the curve has a cusp wheneverφ = (2n+1) π. 4

(4) Consider a pendulum consisting of a mass m hanging from another mass M by astring, where M is only able to move horizontally (i.e., along x), while m can movein the x−z plane under the action of gravity. Write down the equations of motion forboth masses, and then find the frequency of small oscillations. Show that one getsthe expected result in the limitM/m→∞. 2+2+3+1

1

2 PH 201 CLASSICAL MECHANICS FULL MARKS: 85 GIVEN DATE: 15/10/14 DUE DATE: 29/10/14

(5) Find all the frequencies and the corresponding eigenvectors for small oscillations ofthe following systems.

(i) Three particles lie in a straight line, the central particle has mass M , thetwo particles on the sides have mass m each, and there are two springs of length ajoining the central particle with the side particles.

(ii) A tri-atomic molecule in a plane in which the atoms (mass m each) lie at thecorners of an equilateral triangle of side a, and they are joined pair-wise by springs.

In all these systems, assume that the unstretched length of each of the springsis equal to a and the spring constant is k. 5+5

(6) A piston of mass m divides a cubical box of length l into two equal parts; both theparts of the box contain an ideal gas with pressure P . Suppose that the piston isdisplaced slightly to one side by a distance x and then let go. Find the frequencyof the oscillations of the piston, if the process takes place

(i) at constant temperature (PV is a constant),(ii) adiabatically (PV γ is a constant). 3+3

(7) Virial theorem: Consider a Hamiltonian of the form H = T (pi) + V (qi), whereT (pi) is homogeneous in the pi with degree 2, while V (qi) is homogeneous in theqi with degree n. [We say that a function f(xi) is homogeneous in the variablesxi with degree n if f(λxi) = λnf(xi). This implies that

∑i xi∂f/∂xi = nf ]. If

the system is undergoing a motion which is periodic in time, show that the timeaveraged values < T > and < V > satisfy

< T > =n

n+ 2E and < V > =

2

n+ 2E ,

where E = T + V is the energy of the system. [The time averaged value of afunction g(t) is defined to be < g > = limτ→∞ (1/τ)

∫ τ0 dt g(t)].

Hint: Consider the functionG =∑

i piqi. Show that dG/dt = 2T−nV . For periodicmotion, show that < dG/dt >= 0. Then prove the virial theorem. 4+3+2+2

(8) Consider a Lagrangian involving several coordinates, L(qi, qi), and a dissipationfunction F (qi) which is homogeneous in the qi with degree 2. Show that the Hamil-tonian defined as H =

∑i piqi − L satisfies dH/dt = −2F . 4

(9) For a particle moving in a central potential V = α/r2, show that the scatteringangle θs is related to the impact parameter b and the speed at infinity v∞ by

θs = π[

1 − b

(b2 + 2αmv2∞

)1/2

].

Hence show that the differential cross-section is given by

ASSIGNMENT-III 3

dσ

dΩ=

2π2α

mv2∞

π − θsθ2s (2π − θs)2

1

sin θs.

4+4

(10) A particle is moving in a circular orbit of radius r0 in the presence of a centralpotential of the form V (r) = −α/r + γ/r3, where α > 0 and γ is very small. Atsome instant, the particle is suddenly given an impulse in the tangential direction;hence its angular momentum changes by a small amount, but its distance from theorigin remains the same at that instant. Show that the particle then continues tohave a periodic motion in the angular direction as before, but that it undergoessmall oscillations in the radial direction (about a new radius). Compute the ratio ofthe radial and angular frequencies, and use that to compute the rate of precessionof the perihelion to first order in γ.

Your answer should be expressed in terms of α, γ and r0. 3+3+3

(11) A particle of mass m is moving in a circle of radius R in the presence of a centralpotential whose origin lies on the circumference of the circle.

What is the form of the potential V (r)? What is the total energy of the particle,assuming that V (r =∞) = 0? Find the time period of the motion in terms of theorbital angular momentum, R and m. 3+2+3

(12) A particle falls vertically from rest at the equator. After a time t, show that thedeflection from the vertical due to the Coriolis force is given by gωt3/3, where ω isthe angular frequency of the earth’s rotation . If the particle falls from a height of100 m, find the deflection from the vertical when it hits the ground. 2+2

(13) Focault’s pendulum: Consider a simple pendulum suspended vertically over thenorth pole and undergoing small oscillations in the x − y plane. Assume that thependulum’s oscillation frequency ω0 is much larger than the earth’s rotation fre-quency ω, and the pendulum’s length is much smaller than the earth’s radius. Findthe general solution of the equations of motion of the pendulum in a frame fixed tothe earth. Then show that a particular solution corresponds to the pendulum os-cillating along a direction which slowly rotates with the frequency ω. 4+3