Sample Problem: Mountain Climbers

Problem: Suppose five mountain climbers are hanging  from the edge of a cliff, each holding onto the arm of the climber above, forming a chain like that in the figure below (with red circles representing the climbers and the black lines representing their arms).

If each mountain climber (plus his equipment, clothing, etc.) has a weight of 200 pounds, how much force does each mountain climber need to exert?

Answer: The bottom is likely a good place to start, especially since the bottom mountain climber isn't holding up anyone but himself. He thus needs to exert 200 pounds to hold onto the arm of the climber above.

The second-from-the-bottom climber must provide enough upward force to counteract the downward force caused by the bottom climber as well as his own weight. The only downward force caused by the bottom climber is his weight, so the second-from-the-bottom climber must exert a force of 200 + 200 = 400 pounds to hold up the bottom mountain climber.

The middle climber must must provide enough upward force to hold up himself and to counteract the downward force caused by second-from-the-bottom climber, who is providing 200 pounds due to his weight and 200 pounds to support the climber beneath him. Thus, the middle climber must provide 200 + 200 + 200 = 600 pounds of upward force to support the second-from-the-bottom climber.

By extending this reasoning, we see that the next climber up must exert 4*200 pounds = 800 pounds of upward force to support the climbers beneath him, and the top climber must exert 5*200 pounds = 1000 pounds of upward force to support the four climbers beneath him.

The top climber's fingers that are grabbing the ledge must thus provide 1000 pounds of frictional force. We'll talk more about friction next class!