Force, Work, Torque, etc

Exercises

  1. A woman launches a boat from the south shore a straight river that flows directly west at a velocity of four mph. She wants to land her boat at the point on opposite side of river directly opposite from where she launched. If the speed at which she can row (relative to the water) is eight mph, in what direction should she row the boat?

  2. Suppose a paddle-boarder plans on crossing the Colorado river. They are on the south shore at a point where the river is \(200\)ft wide and flowing westward at about 1.5ft/s. They want to travel directly across the river, so they know they must paddle in a direction pointed slightly upstream. Suppose they know they can paddle at about 5ft/s.

    1. In what direction should they paddle?
    2. How long will it take them to cross the river?
  3. Suppose an airplane plans to travel 456 mph with a heading of N30°W. How do you describe this velocity/trajectory as a vector? I.e. what are the east/west and north/south components of the plane’s velocity?
  4. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph. What is the plane’s heading as an angle? What is the plane’s speed?
  5. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but there is a steady 40 mph wind blowing from the southeast. What is the course of the plane as a vector? What is the actual speed of the plane?
  6. Suppose an airplane is flying at a constant speed of 430 mph and has an intended course due east-northeast. However there is a 40 mph wind blowing from the north. What should the plane’s heading be to stay on course?
  7. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but due to the effects of a heavy wind, actually has a course of \(\langle 255, 377 \rangle\) mph. What is the velocity vector of the wind? What is the speed of the wind?
  8. Suppose a 100lb weight is suspended from two cables anchored into the ceiling. One cable makes an angle of 50° with the ceiling while the other cable makes an angle of 32° with the ceiling. What are the tensions (forces) in the cables and the magnitudes of these tensions?

  9. Suppose a 45kg weight is suspended from two cables anchored into the ceiling. One cable is 3m long and the other is 7m long, and they are fastened 9m apart in the ceiling. Nothing is moving.

    1. At what angles are cables hanging relative to the ceiling?
    2. What is the tension (force) along each cable?
  10. Suppose a 12kg cannonball is fired at an initial speed of 180m/s at an angle of inclination of 30°. If the cannon is on level ground and fires over level ground, how far away does the cannonball hit and what’s its speed at impact.
  11. Suppose you’re pushing a 19kg shopping cart. Well, maybe not “pushing,” because you’re just leaning on the handle while you walk along. You exert a force of 300N on the cart at an angle of 40° below horizontal. After pushing the cart 50m, how much work did you do?

  12. Suppose you’re using a 0.2m long wrench to tighten a bolt, and you can apply 43N of force on the wrench handle right at the end of the wrench.

    1. Suppose you apply a force directly orthogonal to the wrench. How much torque do you generate?
    2. Suppose you have a bad grip and can only apply the force at an angle of 70° relative to wrench handle. How much torque do you generate?
    3. Suppose you have a really bad grip, having to reach into a tight space to turn the wrench. You apply the force at an angle of 70° relative to wrench handle, and 20° off the plane in which the bolt should rotate. How much torque do you generate in the direction the bolt tightens?