3 Classical Mechanics - exercise sheet 3
his week’s exercises address rotations, rotary motion, torque and moments of inertia. Please post your
solutions into the box opposite the First Year Labs by 2pm on Friday 12th February.
Reading
Read about rotary motion, torque and moments of inertia in your favourite textbook. e.g.
Fowles & Cassiday Analytical Mechanics (7th ed.)
sections 7.2-7.3, 8.1-8.3, 8.6
Chow
Classical Mechanics (2nd ed.)
sections 12.1-12.4
French & Ebison
Introduction to Classical Mechanics
pp. 255-268
Kibble & Berkshire Classical Mechanics (5th ed.)
sections 9.1-9.2, 9.7
1 Bicycle rims
(5 marks)
A circular loop of radius R = 0.34 m is made from a thin extrusion with a mass per unit length of 0.23 kg m-1.
Calculate its moment of inertia about an axis through its centre and perpendicular to the plane of the loop.
b) A second loop with half the radius (R = 0.17 m) is made from the same extrusion. What is the ratio of the
moments of inertia of the two loops about axes through their centres perpendicular to the planes of the loops?
a)
2 Standard shapes
(5 marks)
a)
Show that the moment of inertia of a uniform right circular cylinder of mass m and radius a about its central
axis is ma2/2.
b) Show that the moment of inertia of a uniform thin spherical shell of mass m and radius a about a diameter is
2ma2/3.
c) Hence show that the moment of inertia of a uniform solid sphere of mass m and radius a about a diameter is
2ma2/5.
3 Icy reception
(5 marks)
Pursued by the most vicious villains of SPECTRE, you find yourself marooned in a pick-up truck on the
annoyingly smooth ice of a frozen lake, upon which the truck’s tyres slide completely freely.
Conveniently, a heavy-duty machine gun is mounted on the back of the truck, which is loaded with plenty
of ammunition. Less conveniently, the gun cannot be trained below the horizontal to strike your foes, so
your only option is to fire the gun over their heads and use the recoil to propel the truck towards sanctuary.
If the gun fires N bullets/shells, each of (ejected) mass m and speed u relative to the truck, and M
(don’t tell her) is the final total mass of the truck and its load, show that, if m/M is so small that you can
treat the problem as approximately continuous, the truck will attain a speed of
Nm 

u ln1 +
.
M 

Explain whether you would have achieved a greater final speed if you had been able to eject the same total
mass Nm at the same relative speed u all at once.
4 Skyfall
(5 marks)
A raindrop of mass m and speed v collects water as it falls through a cloud. Show that, neglecting air resistance,
mg = mv + m v .
Assume that the drop remains spherical and that the rate of accretion is proportional to the cross-sectional area of
the drop multiplied by the speed of fall. Show that this implies that
m = kvm 2 3 ,
for some constant k. Using this equation and the assumption that the drop starts from rest when it is infinitesimally
small, find m as a function of x. Hence show that the acceleration of the raindrop is constant and equal to g/7.
PHYS2006 Classical Mechanics
01/02/2016