Moment of Inertia
- The moment of inertia of an oxygen molecule
about an axis through the centre of mass and perpendicular to
the line joining the atoms is
1.95 × 10-46 kg-m2.
The mass of an oxygen atom is
2.66 × 10-26 kg. What is the distance
between the atoms? Treat the atoms as particles.
- An empty beer can has a mass of 50 g, a length
of 12 cm, and a radius of 3.3 cm. Assume that the shell of the
can is a perfect cylinder of uniform density and thickness.
(a) What is the mass of the lid/bottom?
(b) What is the mass of the shell?
(c) Find the moment of inertia of the can about
the cylinder's axis of symmetry.
- A dumbbell consists of two uniform spheres of
mass M and radius R joined by a thin rod of mass m, length L,
and radius r (see diagram).
(a) What is the moment of inertia about
the centre of mass and perpendicular to the rod (Axis A)?
an axis through one sphere and perpendicular to the rod (Axis
(c) About an axis along the rod (Axis C)?
- A hole of radius r has been drilled in a flat,
circular plate of radius R. The centre of the hole is at a distance
d from the centre of the circle. The mass of the complete body
was M. Find the moment of inertia for rotation about an axis through
the centre of the plate.
- A thick spherical shell has an inner radius R1,
an outer radius R2, and a mass M. The material that the
shell is made of is uniform. Find the moment of inertia of the
thick shell about an axis through the centre of the sphere. The
volume of a sphere is 4πr3/3.
- The object shown in the diagram below consists
of a 100-kg, 25.0 cm radius cylinder connected by four 5.00-kg,
0.75-m long thin rods to a thin-shelled outer cylinder of mass
20.0 kg. A small chunk of metal of mass 1.00 kg has been welded
to the outer cylinder. What is the moment of inertia of the object
about the centre of the inner cylinder? Treat the metal chunk
as a point mass.
- The object in the diagram below consists of five
thin cylinders arranged in a circle. A thin disk has been welded
to the tops and the bottoms of the cylinders. The cylinders each
have a mass of 250 g, a length of 15.0 cm, and a radius of 1.00
cm. The disks are each 125 g and have a radius of 5.00 cm. Find
the moment of inertia of the whole object about an axis through
the centres of the disks.
- Three point masses lying on a flat frictionless surface are
connected by massless rods. Determine the angular acceleration
of the body (a) about an axis through point mass A and out of
the surface and (b) about an axis through point mass B. Express
your answers in terms of F, L, and M. You will need to calculate
the moment of inertia in each case.
- The object in the diagram below is on a fixed frictionless
axle. It has a moment of inertia of I = 50 kg-m2. The
forces acting on the object are F1 = 100 N, F2
= 200 N, and F3 = 250 N acting at different radii R1
= 60 cm, R2 = 42 cm, and R3 = 28 cm. Find
the angular acceleration of the object.
- A rope is wrapped around a solid cylindrical drum. The drum
has a fixed frictionless axle. The mass of the drum is 125 kg
and it has a radius of R = 50.0 cm. The other end of the rope
is tied to a block, M = 10.0 kg. What is the angular acceleration
of the drum? What is the linear acceleration of the block? What
is the tension in the rope? Assume that the rope does not slip.
- Two blocks are connected over a pulley as shown below. The
pulley has mass M and radius R. What is the acceleration of the
blocks and the tension in the rope on either side of the pulley?
(HINT: The tension must be different or the pulley would not rotate.)
- A winch has a moment of inertia of I = 10.0 kg-m2.
Two masses M1 = 4.00 kg and M2 = 2.00 kg
are attached to strings which are wrapped around different parts
of the winch which have radii R1 = 40.0 cm and R2
= 25.0 cm.
(a) How are the accelerations of the two masses and the pulley
(b) Determine the angular acceleration of the masses. Recall that
each object needs a separate free body diagram.
(c) What are the tensions in the strings?
- A rope connecting two blocks is strung over two real pulleys
as shown in the diagram below. Determine the acceleration of the
blocks and angular acceleration of the two pulleys. Block A is
has mass of 10.0 kg. Block B has a mass of 6.00 kg. Pulley 1 is
a solid disk, has a mass of 0.55 kg, and a radius of 0.12 m. Pulley
2 is a ring, has mass 0.28 kg, and a radius of 0.08 m. The rope
does not slip.
- A yo-yo has a mass M, a moment of inertia I, and an inner radius
r. A string is wrapped around the inner cylinder of the yo-yo.
A person ties the string to his finger and releases the yo-yo.
As the yo-yo falls, it does not slip on the string (i.e. the yo-yo
rolls). Find the acceleration of the yo-yo.
- A solid cylinder rolls down an inclined plane without slipping.
The incline makes an angle of 25.0 to the horizontal, the coefficient
of static friction is μs = 0.40, and Icyl
= ½MR2. Hint - you may not assume that static
friction is at its maximum!
(a) Find its acceleration.
(b) Find the angle at which static friction is at its maximum,
at just above this angle the object will start to slip.
- A thin-shelled cylinder rolls up an inclined plane without
slipping. The incline makes an angle of 25.0 to the horizontal,
the coefficient of static friction is μs = 0.40, and
Ihoop = MR2.
(a) Find its acceleration.
(b) Find the angle which the object will start to slip.
- A toy car has a frame of mass M and four wheels of mass m.
The wheels are solid disks. The car is placed on an incline and
let go. Assume each tire supports one-quarter of the car's weight.
(a) Find the acceleration of the toy car.
(b) If the coefficient of static friction is μ, find an expression
for the angle at which the wheels begin to slip.
- A person pulls a heavy lawn roller by the handle with force F so
that it rolls without slipping. The handle is attached to the axle of
the solid cylindrical roller. The handle makes an angle θ to the
horizontal. The roller has a mass of M and a radius R. The coefficients
of friction between the roller and the ground are μs and
(a) Find the acceleration of the roller.
(b) Find the frictional force acting on the roller.
(c) If the person pulls too hard, the roller will slip. Find the value
of F at which this occurs.
- A yo-yo of Mass M, moment of inertia I, and inner and outer radii
r and R, is gently pulled by a string with tension T as shown in the
diagram below. The coefficients of friction between the yo-yo and the
table are μs and μk.
(a) Find the acceleration.
(b) Find the friction acting on the yo-yo.
(c) At what value of T will the yo-yo begin to slip?
Rotational Work and Energy
- A rope is wrapped abound a cylindrical drum as shown
below. It is pulled with a constant tension of 100 N for six
revolutions of the
drum. The drum has a radius of 0.500 m. A brake is also applying a
force to the drum. The brake pushes inwards on the drum with a force of
200 N. The pressure point is 0.350 m from the centre of the drum. The
coefficient of kinetic friction between the brake and the drum is 0.50.
Determine the work done by each torque.
- A rope is wrapped exactly three times around a cylinder with a
fixed axis of rotation at its centre. The cylinder has a mass of 250 kg
and a diameter of 34.0 cm. The rope is pulled with a constant tension
of 12.6 N. The moment of inertia of a cylinder about its centre is I =
½MR2. (a) What is the work down by
the rope as it is
pulled off the cylinder. Note that the rope does not slip. (b) If the
cylinder was initially at rest, what is its final angular velocity?
Note that ropes are always tangential to the surfaces that they are
wrapped around. Note that the work done by the tension is
- A large cylinder of mass M = 150 kg and radius R = 0.350 m.
The axle on which the cylinder rotates is NOT frictionless. A rope is
wrapped around the cylinder exactly ten times. From rest, the rope is
pulled with a constant tension of 25.0 N. The rope does not slip and
when the rope comes free, the cylinder has a forward angular velocity
ωf = 10.5
rad/s. The moment of inertia
of a cylinder is I = ½MR2.
(a) What angle was the cylinder rotated through?
(b) What is the frictional torque of the axle?
(c) How long will it take the frictional torque to bring the cylinder
to a stop?
(d) How many revolutions will it have turned?
- A solid ball, a cylinder, and a hollow ball all have the
same mass m and radius R. They are
allowed to roll down a
hill of height H without slipping. How fast will
each be moving
on the level ground?
- A cylinder of mass M and radius R,
on an incline of angle θ, is attached to a spring of
constant K. The spring is not stretched. Find the
speed of the cylinder when it has rolled a distance L down the incline.
- A block of mass m is connected by a string of negligible
a spring with spring constant K which is in turn fixed to a wall. The
spring is horizontal and the string is hung over a pulley such that the
mass hangs vertically. The pulley is a solid disk of mass M and radius
R. As shown in the diagram below, the spring is initially in its
equilibrium position and the system is not moving.
(a) Determine the speed v of the block after it
has fallen a distance h. Express your answer in terms of g, m, K, M,
(b) The block will oscillate between its initial height and its lowest
point. At its lowest point, it turns around. Use your answer to part
(a) to find where it turns around.
(c) Use your answer to part (a) and calculus to find the height at
which the speed is a maximum.
- A system of weights and pulleys is assembled as shown
pulleys are all fixed. The pulleys on the sides are disks of mass md
and radius Rd; the central pulley is a hoop of
and radius Rh. A massless ideal rope passes
pulleys and joins two weights. The rope does not slip. The two weights
have masses m1 and m2
respectively. Use energy
methods to find the velocity of the weights as a function of
displacement, x. The moment of inertia of a disk is Idisk
½MR2 and of a hoop is Ihoop
- In the diagram block M1 is connected
to M2 by
a very light string running over three identical pulleys. The pulleys
are disks with mass Mp and the rope does not
coefficient of kinetic friction for the horizontal surface that M1
is on is μk. Find an expression
for the speed v of block M2 in terms of the
distance L that
block M1 moves to the right. Your answer should
in terms of L, M1, M2, Mp,
and g only. Hint: work and energy
methods provide the quickest solution.
- A block of mass M on a flat table is connected
by a string of negligible mass to a vertical spring with spring
constant K which is fixed to the floor. The string goes over a pulley
that is a
solid disk of mass M and radius R. As
shown in the
diagram below, the spring is initially in its equilibrium position and
system is not moving. A person pulls the block with force F through a
L. Determine the
speed v of the block
after it has moved distance L. The tabletop is frictionless.
A person on maintains a speed of 20 km/h on an exercise
bike. The radius of the wheel is 20 cm. If the frictional torque on the
wheel is τ = 2.2 N·m, what mechanical power is required to
keep the bike at this speed? If the person is 25% efficient, what power
are they expending?