PHY 3221 Classical Mechanics, Homework #3, due 8:30 am, 02/13/12
1 The escape velocity from the earth is around 11.2 km/s. Getting a rocket to escape can be quite diﬃcult, as this question illustrates. Assume throughout that the rocket under discussion ejects spent fuel with a velocity of 3000 m/s, and neglect all air resistance.
a) Derive an equation giving the velocity of a rocket accelerating vertically against the gravity of the earth, assuming the gravitational force is constant.
b) Show that, even if fuel is discharged at an unlimited rate (kg/s), a rocket must expend fuel comprising more than 97.5% of its initial mass if it is to achieve escape velocity.
c) Suppose that a rocket expends 98% of its initial mass as fuel. What is the maximum time it can take to expend this fuel if it is to achieve escape velocity?
d) What total mass of a rocket is required to launch a 1000 kg payload (assumed to be all that is left after the fuel is spent) so that it just reaches escape velocity 50 s after launch.
e) What constant mass ejection rate in kg/s would be required for the rocket in part (d).
2 This question concerns solid hemispheres and hemispherical shells.
a) Find the center of mass of a solid hemisphere.
b) Consider a body which is hemispherical but not solid: it has an inner radius r
1 and an outer radius r
. Find its center of mass.
c) Suppose that the hemisphere is inﬁnitely thin. Show that the center of mass goes to a limiting position.
d) What must be the thickness (expressed as a fraction of the outer radius) of the hemisphere if its centre of mass is just touching the inner surface? You may need to solve this numerically.
e) Suppose that the hollow hemisphere of part (b) lies above the xy -plane and a solid hemisphere of radius r
1 lies inverted below it, with both centered on the z -axis. What must be the ratio r
2 if the center of mass is to lie exactly at the origin? Assume both parts have ths same density.
3 Suppose that a solid rod of length L and mass M has negligible cross-sectional area.
a) Find the moment of inertia of the rod.
b) Assume that the rod starts out horizontal, but moving vertically, with its center of mass moving upward at speed v
, and that it is rotating about its center of mass with angular frequency ω
c) Find the time at which the rod reaches its maximum height (neglecting air resistance).
d) Suppose that the rod is vertical at the turning point in its orbit. Write down a relation between v
0 and ω
0 in this case.
e) Under the conditions of part (d), ﬁnd the centripetal acceleration for the ends of the rod about its center, assuming that one end of the rod started out with an initial velocity of exactly zero.