10.3. Practice Sets: Work, Energy, and Power

Chapter Equations

[latex]W = F \times d \times \cos\theta[/latex] (1)

[latex]W = \Delta E[/latex] (The work – energy theorem) (2)

[latex]KE = \frac{1}{2} m v^2[/latex] (3)

[latex]PE = m g h[/latex] (4)

[latex]ME_1 = ME_2[/latex] (Conservation of mechanical energy) (5)

[latex]KE_1 + PE_1 = KE_2 + PE_2[/latex] (6)

[latex]\frac{1}{2} m v_1^2 + m g h_1 = \frac{1}{2} m v_2^2 + m g h_2[/latex] (7)

[latex]P = \frac{W}{t} = \frac{\Delta E}{t} = F \times v[/latex] (8)

Practice

1. What force is needed to bring an [latex]850 \, \text{kg}[/latex]car to rest from a speed of [latex]100 \, \text{km/h}[/latex] in a distance of [latex]20 \, \text{m}[/latex]? Calculate from energy consideration.
   Answer:
   [latex]16{,}399 \, \text{N}[/latex]
2. The car going on the roller coaster in the figure has a mass of [latex]370 \, \text{kg}[/latex]. Neglect friction and air resistance.

   10.6. image goes here

   Find:

   1. The potential and the kinetic energy of the car in point A.
      Answer:
      [latex]PE = 36{,}260 \, \text{J}, \quad KE = 740 \, \text{J}[/latex]
   2. The total mechanical energy of the car in point B.
      Answer:
      [latex]ME = 37{,}000 \, \text{J}[/latex]
   3. The speed of the car in point C.
      Answer:
      [latex]v_c = 11.9 \, \text{m/s}[/latex]
   4. The maximum height the car can reach (point D)
      Answer:
      [latex]10.2 \, \text{m}[/latex]
3. A child on a swing being held at a certain height above its lowest position has a total mechanical energy of [latex]4800 \, \text{J}[/latex]. When the swing is released, it reaches the lowest position with a speed of [latex]8.5 \, \text{m/s}[/latex]. Assume no air friction.
   1. Find the total mechanical energy of the system at its lowest position.
      Answer:
      [latex]4800 \, \text{J}[/latex]
   2. Find the mass of the child–swing system.
      Answer:
      [latex]132.9 \, \text{kg}[/latex]
4. A worker is pulling a [latex]12 \, \text{kg}[/latex] crate on the floor, with a rope that makes a [latex]25 \, \text{degrees}[/latex] angle with the horizontal applying a force of [latex]65 \, \text{N}[/latex]. If the worker is moving the crate a distance of [latex]15 \, \text{m}[/latex] in [latex]12 \, \text{s}[/latex], find:
   1. The work done by the worker.
      Answer:
      [latex]884 \, \text{J}[/latex]
   2. The power developed by the worker.
      Answer:
      [latex]74 \, \text{W}[/latex]
5. 1. A [latex]15 \, \text{kg}[/latex] package slides on a level floor at [latex]1.2 \, \text{m/s}[/latex]. Find the work needed to bring the package to a stop. Use an energy approach.
      Answer:
      [latex]-10.8 \, \text{J}[/latex]
   2. If the force applied to the package is [latex]14 \, \text{N}[/latex], what is the distance travelled by the package before it stops?
      Answer:
      [latex]0.77 \, \text{m}[/latex]
6. How long will it take a [latex]1250 \, \text{kg}[/latex] car with a useful power output of [latex]50 \, \text{hp}[/latex] to reach a speed of [latex]20 \, \text{m/s}[/latex] starting from rest?
   Answer:
   [latex]6.7 \, \text{s}[/latex]