**2 Classical Mechanics and Relativity**

2. Stars A and B, of mass mA = 4 × 1031 kg and mB = 12 mA, are orbiting their

common centre of mass on circular orbits. The radius of the orbit of star A is

RA = 107 km.

(a) What is the radius of the orbit of star B? [1]

(b) What is the separation, a, between the stars? [1]

(c) What are the orbital periods of star A and B, respectively? [2]

(d) What are the orbital velocities of star A and B, respectively? [2]

(e) What is the gravitational potential energy of the binary star system? [1]

(f) What is the total kinetic energy? [1]

(g) What is the ratio between the potential and kinetic energy? [1]

(h) What is the total energy of the binary system? [1]

(i) What is the net gravitational acceleration of star B? How does it compare to

the centripetal acceleration needed to keep star B in a circular orbit around

the binary’s centre of mass? [2]

(j) Star A explodes in a supernova. During the supernova, half the mass of

star A is expelled at a velocity of 2 × 104 km s–1 uniformly in all directions

relative to the remnant of the star, which continues to move with the same

velocity as before the supernova explosion. What is the net momentum of

the expelled material? [5]

(k) Use conservation of momentum to calculate the net velocity of the binary

(the velocity of its centre of mass) after the explosion. [2]

(l) What would be the total energy of the binary system in the centre of mass

frame if 75% of the mass of star A was expelled during the supernova? What

happens if the total energy of the binary goes above zero? [6]

**3 Electric Circuits**

3. (a) In the circuit shown below,

answer the following questions.

i. What is the initial battery current immediately after switch, S, is closed?

ii. What is the battery current a long time after switch, S, is closed?

iii. What is the maximum voltage across the capacitor?

iv. If the switch has been closed for a long time and is then opened, deduce

an expression for the current through the 500 kΩ resistor as a function

of time.

v. What is the energy dissipated in the 500 kΩ resistor after the switch is

opened? (Note that 1 kΩ = 103 Ω; 1 MΩ = 106 Ω; 1 µF = 10–6 F.) [12]

(b) i. Use Kirchhoff’s laws to find the current in the 8 Ω resistor in the circuit

shown below. [4]

ii. Repeat the above problem by first finding Thevenin’s equivalent circuit ´

at the terminals A and B. (Hint: To do this you need to remove the 8 Ω

resistor and then find the open circuit voltage VAB and the resistance

looking back into the terminals.) [9]

**4 Electromagnetism**

4. (a) An infinite long wire with a uniform charge density, λ, along its length is

placed along the z-axis. By choosing a suitable Gaussian surface, use

Gauss’ Law to find an expression for the electric field at a distance r from

this wire along the y-axis. [2]

(b) An infinite number of such infinite long wires, each identical to the one described in part (a), are placed in the x–z plane, as shown in the figure below.

All the wires are parallel to the z-axis and equally spaced so that there are

N wires per unit length along the x direction.

Use the superposition principle in combination with the result from (a) to derive an expression for the electric field at a point, with distance d from the

origin, along the y-axis You may find the follow sum useful: |
[10] |

1 X

n=-1

1

a2 + n2 =

π coth (πa)

a

;

where a is any constant.

(c) Show that when d is much greater than the wire spacing, your result from

part (b) approximates to the electric field from an infinite charged plane with

charge per unit area, σ. [3]

(d) A square wire frame with side length a has total resistance R. It is being

pulled with a speed v, perpendicular to the side PQ, out of a region where

there is a uniform magnetic field B pointing out of the page. The extent of

the B field corresponds to the shaded region in the figure below.

Consider the moment when the left corner of the wire frame is a distance x

(≤ a=p2) inside the shaded area.

What is the induced e.m.f. and current in the wire? Express your results in

terms of B, x, v and R. [10]

**5 Mathematics (Term 1)**

5. (a) Given that z = –2+i is a zero of the polynomial z4+5z3+3z2–19z–30,

find the other three zeros. [5]

(b) Find the MacLaurin series for ln (sin x + cos x) up to and including terms

of O(x4). [5]

(c) Find the stationary points of the function f(x) = cos 3x – 6 cos x in the

interval –π=2 ≤ x ≤ π=2 and classify them. You may find useful the

identity sin 3x = 3 sin x – 4 sin3 x. [5]