146
4 Rotational Dynamics
4.44 Consider a point mass m with momentum p rotating at a distance r about an
axis. Starting from the definition of the angular momentum L ≡ r × p of this
point mass, show that
dL
dt
= τ,
where τ is the torque.
A uniform rod of length l and mass M rests on a frictionless horizontal surface.
The rod pivots about a fixed frictionless axis at one end. The rod is initially
at rest. A bullet of mass m travelling parallel to the horizontal surface and
perpendicular to the rod with speed v strikes the rod at its centre and becomes
embedded in it. Using the result above, show that the angular momentum of
the rod after the collision is given by
|L| =
1
2
lv
Is L = (l/2)mv also correct?
What is the final angular speed of the rod?
Assuming M = 5m, what is the ratio of the kinetic energy of the system after
the collision to the kinetic energy of the bullet before the collision?
[University of Durham 2008]
4.45 A uniform sphere of radius r initially at rest rolls without slipping down from
the top of a sphere of radius R. Find the angular velocity of the ball at the
instant it breaks off the sphere and show that the angle θ = cos −1 (10/17)
with the vertical.
4.46 A uniform rod of mass m and length 2a is placed vertically with one end in
contact with a smooth horizontal floor. When it is given a small displacement,
it falls. Show that when the rod is about to strike, the reaction is equal to mg/4.
[courtesy from R.W. Norris and W. Seymour, Mechanics via Calculus,
Longmans & Co.]
4.47 The double pulley shown in Fig. 4.12 consists of two wheels which are fixed
together and turn at the same rate on a frictionless axle. A rope connected
to mass m 1 is wound round the circumference of the larger wheel and a
second rope connected to mass m 2 is wound round the circumference of the
smaller wheel. Both ropes are of negligible mass. The moment of inertia, I ,
of the double pulley is 38 kg m
2 . The radii of the wheels are R 1 = 1.2 m and
R 2 = 0.5 m.
(a) If m 1 = 25 kg, what should the value of m 2 be so that there is no angular
acceleration of the double pulley?
(b) The mass m 1 is now increased to 35 kg and the system released from rest.
4 Rotational Dynamics
4.44 Consider a point mass m with momentum p rotating at a distance r about an
axis. Starting from the definition of the angular momentum L ≡ r × p of this
point mass, show that
dL
dt
= τ,
where τ is the torque.
A uniform rod of length l and mass M rests on a frictionless horizontal surface.
The rod pivots about a fixed frictionless axis at one end. The rod is initially
at rest. A bullet of mass m travelling parallel to the horizontal surface and
perpendicular to the rod with speed v strikes the rod at its centre and becomes
embedded in it. Using the result above, show that the angular momentum of
the rod after the collision is given by
|L| =
1
2
lv
Is L = (l/2)mv also correct?
What is the final angular speed of the rod?
Assuming M = 5m, what is the ratio of the kinetic energy of the system after
the collision to the kinetic energy of the bullet before the collision?
[University of Durham 2008]
4.45 A uniform sphere of radius r initially at rest rolls without slipping down from
the top of a sphere of radius R. Find the angular velocity of the ball at the
instant it breaks off the sphere and show that the angle θ = cos −1 (10/17)
with the vertical.
4.46 A uniform rod of mass m and length 2a is placed vertically with one end in
contact with a smooth horizontal floor. When it is given a small displacement,
it falls. Show that when the rod is about to strike, the reaction is equal to mg/4.
[courtesy from R.W. Norris and W. Seymour, Mechanics via Calculus,
Longmans & Co.]
4.47 The double pulley shown in Fig. 4.12 consists of two wheels which are fixed
together and turn at the same rate on a frictionless axle. A rope connected
to mass m 1 is wound round the circumference of the larger wheel and a
second rope connected to mass m 2 is wound round the circumference of the
smaller wheel. Both ropes are of negligible mass. The moment of inertia, I ,
of the double pulley is 38 kg m
2 . The radii of the wheels are R 1 = 1.2 m and
R 2 = 0.5 m.
(a) If m 1 = 25 kg, what should the value of m 2 be so that there is no angular
acceleration of the double pulley?
(b) The mass m 1 is now increased to 35 kg and the system released from rest.