A team studying trees in a rain forest builds a rope bridge between two trees by passing a rope over two tree branches and attaching equal mass counterweights to each end of the rope. When a team member walks along the rope bridge, the bridge sags and the counterweights raise some. Assume that the tree branches are frictionless. A team member stops halfway between the trees.
Calculate how far the bridge is sagging (h) in terms of the mass of the team member (M), the mass of the counterweights (m), and the distance the team member is from one of the trees (d).
(a) What variable are you trying to solve for in this problem? What variables can you treat as known in this problem, i. e. what variables may be in your final answer?
(b) Describe the physics of what is happening in this situation. Are any of the objects moving? If so, what is their velocity? Their acceleration? Are there any forces being applied to any object? Are the forces balanced or unbalanced?
(c) Is the main principle in this problem kinematics (motion, velocity, acceleration from chapters 2-3) or forces?
(d) Draw a free body diagram for object B, including vectors for all of the forces acting on object B. Draw a free body diagram for object A. The force vectors should be in the correct directions and to approximately the correct scale. Label each force vector. Are any of your force
vectors equal in magnitude?
(e) Choose a coordinate system and draw the axes near your free body diagrams.
(f) For each of the force vectors on your free body diagrams, find the x and y components of the force.
(g) Are the horizontal forces on object B balanced? Are the vertical forces on object B balanced? Is the sag in the rope determined by the horizontal or vertical forces?
(h) Write down the general equation for Newton’s 2nd law. Apply Newton’s 2nd law to the vertical components of the forces on object B. This should give you an equation that relates the masses of all the objects and the angle of the rope.
(i) To relate the angle of the rope to the distances d and h, draw a triangle with the hypotenuse from object B to one of the pulleys, base d, and height h. Use trigonometry to get a relationship between the angle of the rope, d and h.
(j) You now have 2 equations from parts (h) and (i). Count how many unknowns are in
these equations. Do you have enough information to solve the problem?
(k) Use algebra to solve your equations from (h) and (i) for the height h in terms of the masses m and M and the distance d, eliminating the unwanted variable for the angle of the rope.
(l) Check that your equation has correct units. If the mass of the counterweights increases, what happens to the amount of sagging? If the mass of object B increases, what happens to the amount of sagging?
(m) Sketch a graph of the amount of sagging (h) vs. the mass of object B (M). What happens when M is very small? What happens when M is very large?