PHY218 SPRING 2016 Review for Exam#3: Week 12 Review: Linear Momentum, Collisions, Rotational Motion, and Equilibrium

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1 PHY218 SPRING 2016 Review for Exam#3: Week 12 Review: Linear Momentum, Collisions, Rotational Motion, and Equilibrium These are selected problems that you are to solve independently or in a team of 2-3 in order to better prepare for your Exam#2 1

2 Problem 1: The Ballistic Pendulum A block of mass M is suspended by a massless wire of length L and, after a completely inelastic collision with a bullet of mass m, swings up to a maximum height y < L. Find: (a) the speed v of the block with the bullet in it immediately after impact; (b) the tension force T in the wire immediately after impact; (c) the initial horizontal velocity v o of the bullet. Problem 2: The center of mass A long thin rod of length L has a linear density λ(x) = Ax, where x is the distance from the left end of the rod. (a) What is the mass of the rod? (b) How far is the center of mass of the rod from the left end of the rod? 2

3 Problem 3: Head-on elastic collision Find velocities v 1 and v 2 of two balls of masses m 1 and m 2, respectively, after their elastic head-on (one-dimensional) collision, if, before the collision, the first ball had velocity v 10 and the second ball was at rest. Problem 4: Lost wheel A wheel of mass M and radius R comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is moving at 10 m/s. Its moment of inertia about its rotation axis is 0.8 MR 2. The wheel rolls up the hill to a stop, a height h above the bottom of the hill. An irreversible frictional loss of energy on this path is equal to one third of the total kinetic energy that the wheel had at the bottom of the hill. (a) Calculate h. (b) What would be the height h for a wheel in the form of a solid homogeneous disk? 3

4 Problem 5: Neutron Star Glitches Occasionally, a rotating neutron star undergoes a sudden speedup called a glitch. It occurs when the crust of the neutron star settles slightly, decreasing the moment of inertia about the rotation axis. Suppose an angular speed of the star increased from ω o to ω = ω o +Δω, where Δω/ω o = If the radius of the neutron star before the glitch was Ro = 10 km, by how much did its radius decrease in the starquake? Assume that the neutron star is a uniform sphere. Problem 6: Rolling downward an incline Consider a symmetric body rolling without slipping down an incline that makes an angle β with a horizontal. The body starts from rest at a height h above the bottom of the incline and has a radius R, a mass M, and a moment of inertia I=cMR 2 about an axis of rotation through its center of mass, where c is a constant, 0 < c < 1. Find: (a) the body s speed v, (b) the body s acceleration a (c) the time of travel to the bottom of the incline. 4

5 Problem 7: Blocks descending over a massive pulley In the diagram below, the pulley has a radius R and a moment of inertia I. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between the block and the tabletop is µk. The block on the tabletop has mass m 1. The descending block has mass m 2. The system is released from rest. Find: (a) The speed v and the acceleration a of the blocks when the distance of descent is d (b) The tension forces which the rope exerts on the blocks. (c) Which of the two tensions is larger? Problem 8: Raising a crate via a rope wrapped around a cylinder that turns on a metal axle attached to a crank handle The mechanism shown below is used to raise a crate of supplies from a ship s hold. The crate has mass M. The cylinder has radius R and moment of inertia I. The crate is suspended from the free end of the rope. The cylinder pivots on frictionless bearings. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius r, the cylinder turns, and the crate is raised. What magnitude of the force F applied tangentially to the rotating crank is required to raise the crate with an acceleration a? r 5

6 Problem 9: Marbles in a container Two uniform marbles of the same mass m and radius R are stacked in a container that is 3R wide. All surfaces are frictionless. (a) Find the force that the container exerts on the marbles at the points of contact A, B, and C. (b) What force does each marble exert on the other? Problem 10: Bicycle wheel and a curb You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this, you apply a horizontal force F as shown in figure. What is the smallest magnitude of the force F that will succeed in raising the wheel onto the curb when the force is applied (a) at the center of the wheel and (b) at the top of the wheel? (c) In which case is less force required? 6

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