4.2 Problems
139
melted and recasted into a solid sphere. Show that the moment of inertia of the
sphere about its diameter is I/5.
4.10 Calculate the moment of inertia of a hollow sphere of mass M and radius R
about its diameter.
4.11 Use the formula for moment of inertia of a uniform sphere about its diameter
I =
2
5
MR 2
to deduce the moment of inertia of a thin hollow sphere about
the axis passing through the centre.
4.2.2 Rotational Motion
4.12 A solid cylinder of mass m and radius R rolls down an inclined plane of height
h without slipping. Find the speed of its centre of mass when the cylinder
reaches the bottom.
4.13 A star has initially a radius of 6 × 10 8 m and a period of rotation about its
axis of 30 days. Eventually it evolves into a neutron star with a radius of only
10 4 m and a period of 0.1 s. Assuming that the mass has not changed, find the
ratio of initial and final (a) angular momentum and (b) kinetic energy.
4.14 A uniform solid ball rolls down a slope. If the ball has a diameter of 0.5 m and
a mass of 0.1 kg, find the following:
(a) The equation which describes the velocity of the ball at any time, given
that it starts from rest. Clearly state any assumptions you make.
(b) If the slope has an incline of 30 ◦ to the horizontal, what is the speed of
the ball after it travels 3 m?
(c) At this point, what is the angular momentum of the ball?
(d) If the coefficient of friction between the ball and the slope is 0.26, what is
the maximum angle of inclination the slope could have which still allows
the ball to roll?
[University of Durham 2000]
4.15 (a) Show that the least coefficient of friction for an inclined plane of angle θ in
order that a solid cylinder will roll down without slipping is
1
3
tan θ . (b) Show
that for a hoop the least coefficient of friction is
2
3
tan θ .
4.16 A small mass m tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is drawn through a hole with constant
velocity, Fig. 4.2. Show that the tension in the thread is inversely proportional
to the cube of the distance from the hole.
[Osmania University]
139
melted and recasted into a solid sphere. Show that the moment of inertia of the
sphere about its diameter is I/5.
4.10 Calculate the moment of inertia of a hollow sphere of mass M and radius R
about its diameter.
4.11 Use the formula for moment of inertia of a uniform sphere about its diameter
I =
2
5
MR 2
to deduce the moment of inertia of a thin hollow sphere about
the axis passing through the centre.
4.2.2 Rotational Motion
4.12 A solid cylinder of mass m and radius R rolls down an inclined plane of height
h without slipping. Find the speed of its centre of mass when the cylinder
reaches the bottom.
4.13 A star has initially a radius of 6 × 10 8 m and a period of rotation about its
axis of 30 days. Eventually it evolves into a neutron star with a radius of only
10 4 m and a period of 0.1 s. Assuming that the mass has not changed, find the
ratio of initial and final (a) angular momentum and (b) kinetic energy.
4.14 A uniform solid ball rolls down a slope. If the ball has a diameter of 0.5 m and
a mass of 0.1 kg, find the following:
(a) The equation which describes the velocity of the ball at any time, given
that it starts from rest. Clearly state any assumptions you make.
(b) If the slope has an incline of 30 ◦ to the horizontal, what is the speed of
the ball after it travels 3 m?
(c) At this point, what is the angular momentum of the ball?
(d) If the coefficient of friction between the ball and the slope is 0.26, what is
the maximum angle of inclination the slope could have which still allows
the ball to roll?
[University of Durham 2000]
4.15 (a) Show that the least coefficient of friction for an inclined plane of angle θ in
order that a solid cylinder will roll down without slipping is
1
3
tan θ . (b) Show
that for a hoop the least coefficient of friction is
2
3
tan θ .
4.16 A small mass m tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is drawn through a hole with constant
velocity, Fig. 4.2. Show that the tension in the thread is inversely proportional
to the cube of the distance from the hole.
[Osmania University]