1. Though it has no true springs in it, the bridge behaves like one. In effect, you are standing motionless on a huge spring. Briefly explain why you are now at a stable equilibrium.

Answer: You are not accelerating, so you are experience zero net force and are therefore at equilibrium. If you are disturbed from equilibrium, you will experience a restoring force back toward that equilibrium, so the equilibrium is stable.

Why: You are essentially a mass hanging on a spring, resting motionless at the equilibrium point. The upward force exerted on you by the spring is exactly balancing your downward weight and you are experiencing zero net force. If you're disturbed up or down, the restoring force will change and you'll accelerate back toward equilibrium.

2. Before you walked to the middle of the bridge, its motionless surface was 5 feet higher than it is now. Your friend, who weighs the same amount as you, now walks slowly onto the bridge from one end and joins you in the middle. The bridge is again motionless and the two of you are at a stable equilibrium. Has the bridge's surface moved up or down and, if so, by how much?

Answer: The bridge's suface has moved down by an additional 5 feet (a total of 10 feet overall).

Why: To support the doubled weight, the spring must exert twice its original upward force. To develop that much upward force, it must be stretched downward twice as far as before, from the original 5 feet to the current 10 feet down.

3. Because you are holding the railings so tightly, you are bouncing with the bridge and never leave its surface. At what moment(s) during a single complete bounce (up and down),

(A) are you traveling upward fastest,
(B) are you traveling downward fastest,
(C) are you accelerating upward fastest,
(D) are you accelerating downward fastest, and
(E) do you have the greatest kinetic energy?

Answer:

(A) as you pass upward through the equilibrium point.
(B) as you pass downward through the equilibrium point.
(C) as you reach the lowest point in your bouncing motion.
(D) as you reach the highest point in your bouncing motion.
(E) as you pass through the equilibrium point in either direction.

Why: You reach your maximum speed when you stop accelerating forward and just before you begin decellerating. That situation occurs each time you pass through equilibrium. You also have you maximum kinetic energy then, because your kinetic energy is proportional to the square of your speed. You reach your maximum acceleration when the force you are experiencing hits its maximum. That situation occurs each time your reach your maximum distance from equilibrium and the balance between weight and spring force is at its worst.

4. The bouncing gradually diminishes and you are again motionless at the middle of the bridge. You release your grip on the rope railings just as your cheerful friend slaps you hard on the back. Use the concept of an impulse to prove that this slap transfers horizontal momentum from your friend to you.

Answer: Your friend pushes you horizontally for an amount of time, which is an impulse. You therefore receive horizontal momentum from your friend.

Why: Whenever something pushes on you for more than an instant, it gives you an impulse and you receive momentum as a result. What you do with that momentum is your business... you can give it to the ground or you can begin to move with it.

5. Your new momentum carries you off the side of the bridge and you suddenly having nothing under your feet. Down you go! As you fall, how do the following quantities change (or not change):

(A) your gravitational potential energy,
(B) your kinetic energy, and
(C) your total energy?

(Assume no air resistance and that you don’t touch anything as you fall.)

Answer: (A) your gravitational energy decreases, (B) your kinetic energy increases, and (C) your total energy remains constant.

Why: Since you can't do work on anything as you fall, your energy remains in you. However, it can change forms and goes from gravitational potential at your highest position to kinetic energy at your lowest position.

6. As you fall feet-first, you notice that you are spinning about a vertical axis like a toy top. You try to stop spinning, to get a better view of what lies below you, but nothing you do seems to stop your spin. Use the concept of angular momentum to explain briefly why you can’t stop spinning but why you can slow your rate of rotation by extending your arms out horizontally.

Answer: Since you cannot exchange angular momentum while you fall, you are stuck with whatever angular momentum you have. Since you are spinning, you must continue spinning. But by increasing your angular mass (or rotational mass, moment of inertia, or rotational inertia), you can make your angular velocity decrease.

Why: Your angular momentum can't change as you fall, but you can change the relationship between angular mass and angular velocity. By increasing your angular mass, you decrease your angular velocity. The product of those two,: your angular mass times your angular velocity, is your angular momentum and that can't change.

7. Below you are two possible targets: deep water and a wide, flat rock. Hitting either one will stop your downward motion, but the water will stop you more comfortably than the rock. Use the concepts of momentum and impulse to explain why it would feel better to hit the water than to hit the rock.

Answer: You will transfer all your downward momentum to either object, but the impulse involved when hitting the water will be a smaller force for a longer time. The smaller force feels better.

Why: You give the same impulse to the water and to the rock, but the water impulse involves a small force exerted over a long time. Had you hit the rock, you would have lost your downward momentum painfully quickly, via a huge force exerted over a short time.

8. Luckily, you hit the water and float safely back to the surface. After swimming to shore, you scramble up the river bank to one end of the bridge. Together with some other members of your group, you grab hold of that end of the bridge and pull on it horizontally as hard as you can. The bridge suddenly pulls taut: it stops drooping in the middle and becomes almost perfectly straight. As the bridge rises toward horizontal, your friend, who was still in the middle of the bridge, is tossed about 20 feet into the air above the bridge. Where did the energy that lifted your friend upward come from? Use the concept of work to prove that energy was transferred first from you to the bridge and second from the bridge to your friend?

Answer: The energy can from you (pulling on the bridge). You did work on the end of the bridge as you pulled it toward you and it moved toward you. The bridge then did work on your friend as it pushed your friend upward as your friend moved upward.

Why: Energy flows from you to the bridge to your friend. In each case work was done: a force exerted on the object that receives energy and that object moves in the direction of the force.