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Physics 1120 In-Class Problems: Work & Energy

Energy

  1. In the diagram below, the spring has a force constant of 5000 N/m, the block has a mass of 6.20 kg, and the height h of the hill is 5.25 m. Determine the compression of the spring such that the block just makes it to the top of the hill. Assume that there are no non-conservative forces involved.

  2. 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?

  3. 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.

  4. 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) Use energy methods, to 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.

  5. 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 md and radius Rd; the central pulley is a hoop of mass mh and radius Rh. 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 Idisk = ½MR2 and of a hoop is Ihoop = MR2.


  6. In the diagram below, a moving skier on top of a circular hill of radius Rh = 62.0 m feels that his "weight" is only (3/8)mg. What would be the skier's apparent weight (in multiples of mg) at the bottom of the circular valley which has a radius Rv = 43.0 m? Neglect friction and air resistance.

  7. At point A in the figure shown below, a spring (spring constant k = 1000 N/m) is compressed 50.0 cm by a 2.00 kg block. When released the block travels over the frictionless track until it is launched into the air at point B. It lands at point C. The inclined part of the track makes an angle of θ = 55.0° with the horizontal and point B is a height h = 4.50 m above the ground. How far horizontally is point C from point B?

  8. A small solid sphere of radius r = 1.00 cm and mass m = 0.100 kg at point A is pressed against a spring and is released from rest with the spring compressed 20.0 cm from its natural length. The spring has a force constant k = 20.0 N/m. The sphere rolls without slipping along a horizontal surface to point B where it smoothly continues onto a circular track of radius R = 2.00 m. The ball finally leaves the surface of the track at point C. Find the angle θ where the ball leaves the track. Assume that friction does no work. Hint - find an expression for the speed of the sphere at point C. The moment of inertia of a sphere is I = 2/5mr2.

  9. The potential energy of a system of particles in one dimension is given by:

    ,

    where the potential energy is in Joules. What is the work done in moving a particle in this potential from x = 1 m to x = 2 m? What is the force on a particle in this potential at x = 1 m and at x = 2 m? Where are the points of stable and unstable equilibrium (peaks and troughs)?

  10. The PE curve for a 2.5-kg particle under the influence of a conservative force is shown below. (a) What would be the total mechanical energy of the system, if one know that the particle has a speed of 7.5 m/s at x = 1.5 m? (b) If Etot = 150 J, where would the particle have zero velocity? Where would the particle have its maximum kinetic energy and what would be its speed there? (c) What is the minimum total energy for which the particle escapes the influence of the force creating the potential (i.e. the particle escapes the potential well)?

Work and Energy

  1. In the diagram below, calculate the work done if:
    (a) F = 15.0 N, θ = 15°, and Δx = 2.50 m,
    (b) F = 25.0 N, θ = 75°, and Δx = 12.0 m,
    (c) F = 10.0 N, θ = 135°, and Δx = 5.50 m,

  2. Determine the work done by the following. Determine the angles between the forces and the displacements. The forces are in Newtons and the displacements are in metres:
    (a) F = <1, 2, 3> and Δr = <4, 5, 6>
    (b) F = <1, 2, 3> and Δr = <4, 5, -6>
    (c) F = <4, 2, 4> and Δr = <2, -8, 2> 

  3. In the diagram below, a rope with tension T = 150 N pulls a 15.0-kg block 3.0 m up an incline (θ = 25.0°). The coefficient of kinetic friction is μk = 0.20. Find the work done by each force acting on the block.

  4. A winch lifts a 150 kg crate 3.0 m upwards with an acceleration of 0.50 m/s2. How much work is done by the winch? How much work is done by gravity?

  5. What work does a baseball bat do on a baseball of mass 0.325 kg which has a forward sped of 36 m/s and a final speed of 27 m/s backwards. Assume motion is horizontal.

  6. What is the work done by friction in slowing a 10.5-kg block traveling at 5.85 m/s to a complete stop in a distance of 9.65 m? What is the kinetic coefficient of friction?

  7. A 50.0-N force is applied horizontally to a 12.0-kg block which is initially at rest. After traveling 6.45 m, the speed of the block is 5.90 m/s. What is the coefficient of kinetic friction?

  8. 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.

  9. In the diagram below, a 5.00-kg block slides from rest at a height of h1 = 1.75 m down to a horizontal surface where it passes over a 2.00 m rough patch. The rough patch has a coefficient of kinetic friction μk = 0.25. What height, h2, does the block reach on the incline?

  10. In the diagram below, a 5.00-kg block slides from rest at a height of h1 = 1.75 m down to a smooth horizontal surface until it encounters a rough incline. The incline has a coefficient of kinetic friction μk = 0.25. What height, h2, does the block reach on the θ = 30.0° incline?

  11. 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.

  12. Suppose that there is friction in problem 1 and that the compression must in fact be 0.425 m for the block to just reach the top of the hill. What work is done by the frictional force?

  13. 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?

  14. A person with an axe to grind is using a whetstone. The whetstone is connected to a motor which keeps it rotating at 85 rev/min. The whetstone is a solid cylinder of made of a special type of stone of mass 45 kg and radius 20 cm. The heavy axe-blade is being pressed onto the whetstone with a steady force of 25 N directed into the centre of the whetstone. Suddenly, the power to the motor is cut but the person maintains the force of the axe on the whetstone. The coefficient of kinetic friction between the axe and the stone is 0.60. The moment of inertia of a cylinder about it's centre of mass is I = ½MR2.
    (a) Calculate the torque of the frictional force on the wheel.
    (b) Calculate the total angle through which the wheel turns from the time the power goes off to the time the whetstone stops rotating.
    (c) How long does it take for the wheel to stop after the power outage?

  15. In the figure below, a block of mass 5.0 kg starts at point A with a speed of 15.0 m/s on a flat frictionless surface. At point B, it encounters an incline with coefficient of kinetic friction μk = 0.15. The block makes it up the incline to a second flat frictionless surface. What is the work done by friction? What is the velocity of the block at point C? The incline is 2.2 m long at an angle θ = 15°.
  16. 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 slip. The 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 be expressed in terms of L, M1, M2, Mp, μk, and g only. Hint: work and energy methods provide the quickest solution.
  17. Tarzan, Lord of Apes, is swinging through the jungle. In the diagram below, Tarzan is standing at point A on a tree branch h1 = 22.0 m above the floor of the jungle. Tarzan is holding one end of a vine which is attached to a branch on a second tree. The vine is L = 21.0 m long. When Tarzan swings on the vine, his path is in an arc of a circle. At the bottom of his swing he is at point B, 13.0 m above the ground . Ignore Tarzan's height. Tarzan has a mass of 90.0 kg. The vine does not stretch and has negligible mass.
    (a) Why does the tension in the vine do no work?
    (b) What will be his speed at point B?
    (c) What will be the tension in the rope at point B?

  18. 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.

  19. What power is required to pull a 5.0 kg block at a steady speed of 1.25 m/s? The coefficient of friction is 0.30.

  20. A 7500 W engine is propelling a boat at 12 km/h. What force is the engine exerting on the boat? What force and how much power is water resistance exerting on the speedboat?

  21. A 3.0 hp engine pulls a 245-kg block at constant speed up a 12.0 m 30.0° incline. How long does this take? Ignore friction.

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