At height H above the ground, a hollow sphere of mass m and radius r is released from rest from the top of a hill. There is a bump that is 2H/3 above the ground. Assume the sphere moves without any slipping. Assume air resistance is negligible, and gravity has acceleration of g. Which physical quantities are conserved over the course of the displacement? What is the linear speed of the ball at the lowest point? What fraction of the sphere's mechanical energy is rotational kinetic energy when the sphere reaches the top of the bump? Explain how a block sliding without friction would have a larger/equal/smaller linear speed at the end of the track, without using any equations or calculations.

Engineers are designing a system by which a falling mass m imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (Figure 1). There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is 3.71 m/s?. In the earth tests, when m is set to 15.0 kg and allowed to fall through 3.00 m, it gives 350.0 J of kinetic energy to the drum. Part A If the system is operated on Mars, through what distance would the 15.0 kg mass have to fall to give the same amount of kinetic energy to the drum? Express your answer with the appropriate units. HA hM = Value Units Part B Figure 1 of 1 How fast would the 15.0 kg mass be moving on Mars just as the drum gained 350.0 J of kinetic energy? Express your answer with the appropriate units. Drum HA Value Units v =

5 Problem For the pulley system show below. Determine an expression for the velocity VA = SA of the cart A in terms of the velocity VB = Śg of the mass B. Hint: you can lump all constants related to the length of the rope into a single constant k that vanishes when differentiating. SA B SB

Given a moving cart (0) with a pivoting bar. The cart is at a distance from the fixed wall and the particle P is at y distance (y # 0) from the bar's pivot point. Assume planar motion. X (a) Define an inertial frame e and label all necessary quantities on the plot that you will be using for finding P's position, velocity and acceleration. You are free to define any additional frames. (b) Write expression for P's position in e frame. What is your choice for the base point of P's position vector if we are interested in obtaining its velocity and acceleration ex- pression? (c) Find P's velocity and acceleration in e frame, i.e. 7² and ².