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?

(b) What is the separation, a, between the stars? 
(c) What are the orbital periods of star A and B, respectively? 
(d) What are the orbital velocities of star A and B, respectively? 
(e) What is the gravitational potential energy of the binary star system? 
(f) What is the total kinetic energy? 
(g) What is the ratio between the potential and kinetic energy? 
(h) What is the total energy of the binary system? 
(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?

(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 s1 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?

(k) Use conservation of momentum to calculate the net velocity of the binary
(the velocity of its centre of mass) after the explosion.

(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?


3 Electric Circuits
3. (a) In the circuit shown below,
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 k
resistor as a function
of time.
v. What is the energy dissipated in the 500 k
resistor after the switch is
opened? (Note that 1 k
= 103 ; 1 M= 106 ; 1 µF = 106 F.) 
(b) i. Use Kirchhoff’s laws to find the current in the 8 resistor in the circuit
shown below.

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
resistor and then find the open circuit voltage VAB and the resistance
looking back into the terminals.)


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. 
(b) An infinite number of such infinite long wires, each identical to the one described in part (a), are placed in the xz 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: 

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. 
(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
What is the induced e.m.f. and current in the wire? Express your results in
terms of
B, x, v and R. 

5 Mathematics (Term 1)
5. (a) Given that z = 2+i is a zero of the polynomial z4+5z3+3z219z30,
find the other three zeros.

(b) Find the MacLaurin series for ln (sin x + cos x) up to and including terms
of
O(x4). 
(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. 

Physics coursework