Moment of Inertia


1. 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.

2. 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.
   

3. 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)?
   (b) About an axis through one sphere and perpendicular to the rod (Axis B)?
   (c) About an axis along the rod (Axis C)?
   
   

4. 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.
   

5. A thick spherical shell has an inner radius R₁, an outer radius R₂, and a mass M. The material that the spherical 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.
   

6. 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.
   

7. 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.
   
   

   Pulleys
   

8. 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.
   

9. 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.
   

10. 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.
    

11. 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.)
    

12. 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 related?
    (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?
    
    

13. 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.
    
    

    Rolling Objects
    

14. 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.
    
    

15. 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 μ_s = 0.40, and I_cyl = 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.
    

16. 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 μ_s = 0.40, and I_hoop = MR².
    (a) Find its acceleration.
    (b) Find the angle which the object will start to slip.
    

17. 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.
    
    

18. 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 μ_s and μ_k.
    (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.
    

    

19. 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 μ_s and μ_k.
    (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
    

20. 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.
    

21. 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 non-conservative.
    

22. 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 ω_f = 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?
    

23. 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?

    

24. 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.

    

25. A block of mass m is connected by a string of negligible mass to 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, and h.
    (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.
    
26. A system of weights and pulleys is assembled as shown below. The pulleys are all fixed. The pulleys on the sides are disks of mass m_d and radius R_d; the central pulley is a hoop of mass m_h and radius R_h. A massless ideal rope passes around the 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 I_disk = ½MR² and of a hoop is I_hoop = MR².
    

27. 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 M_p and the rope does not slip. The coefficient of kinetic friction for the horizontal surface that M₁ is on is μ_k. 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 be expressed in terms of L, M₁, M₂, M_p, μk, and g only. Hint: work and energy methods provide the quickest solution.
    

28. 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 the system is not moving. A person pulls the block with force F through a distance L. Determine the speed v of the block after it has moved distance L. The tabletop is frictionless.
    

29. 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?  

Questions? mikec@kwantlen.bc.ca