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Please answer all of them, Thank you.

1. What are the three equivalent definitions of a conservativeforce? Show how they are equivalent.

2. Suppose there is a pendulum with length 5m hanging from aceiling. A ball of mass 2kg is attached to the bottom of thependulum. The ball begins at rest. If I give the ball a velocityof
6 m/s, what is the maximum height that the ball will achieve? Usethe energy conservation model to solve, and assume that there is nofriction or air resistance.

3. I hang a (massless) spring from the ceiling. When the springis lying horizontally, it has equilibrium length of 0.75m, but alonger equilibrium length once it is hanging from the ceiling witha mass attached (see previous homework). I then attach a block withmass 0.5kg to the bottom of the spring and let the system come toequilibrium. If the block starts at rest, how much speed must Igive it so that strikes the ceiling? How far will it stretch thespring when it bounces back?

4. For the following situations given the system described,which of the quantities listed should be zero? For those that arenot zero, provide its sign. (External work, change in kineticenergy, change in gravitational potential energy, change in elasticpotential energy, change in thermal energy)

a. System: Ice skater. Situation: (Initial state = Ice skater atrest on ice, final state: ice skater at high speed on the ice)

b. System: Ice skater + ice rink. Situation: (Initial state =Ice skater at rest on ice, final state: ice skater at high speed onthe ice)

c. System: Earth + comet. Situation: (Initial state = comet isfar from earth, final state = comet has fallen to Earth, rightbefore it impacts the ground)

d. System: Earth + hanging spring with mass attached. Situation:(Initial state = mass at rest at equilibrium position, final state= mass at rest with spring fully compressed)