Assignment 1

Problem 1:

A rugby player runs with the ball directly towards his opponent's goal, along the positive direction of an x axis.  He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive x component.  Suppose the player runs at speed 4.0 m/s relative to the field while he passes the ball with a speed of 6.0 m/s relative to himself.  What is the smallest angle (relative to the x axis) the ball can be passed in (as seen from the player) in order for the pass to be legal?

Solution:

Problem 2:

A 10 g bullet is stopped in a block of wood (m = 5 kg).  The speed of the bullet-wood combination immediately after the collision is 0.6 m/s.  What was the original speed of the bullet?

Solution:

Problem 3:

imageAn object is sliding on a frictionless surface with velocity V0.  There is a track on the surface, which turns object backwards as shown by the dashed line in the figure.  If the radius of the track is R and the coefficient of kinetic friction between object and track is μ, how long will it take for the object to make a 180 degree turn?
The size of the object is negligible compared to the radius of the track.

Solution:

Problem 4:

A small massive bead can slide without friction along a wire ring of radius R mounted in the vertical plane.  A light elastic cord is attached to the top of the ring and to the bead.  The relaxed length of the cord is L.  If the cord is stretched by ΔL > 0, it exerts an elastic force F = -kΔL directed toward the attachment point on top of the ring.
Initially, the bead is at rest at the bottom of the ring, and the magnitude of the force exerted on the bead by the ring is twice the bead's weight.  After a small disturbance, the bead begins to slide upward along the ring, reaching its maximum velocity at the moment it covers one-third of the ring's circumference. 
(a)  What is the length of the relaxed cord?
(b)  What is vmax?
(c)  Find the force exerted by the ring on the bead when v = vmax.
 

image

Solution:

Problem 5:

The density of aluminum is 2.7 g/cm3.  A 10 cm diameter aluminum cable is used to lower a 1000 kg mass into a 300 m deep bore hole.
(a)  When the mass is held just above the bottom of the hole, what is the tension in the cable as a function of depth?  At what depth is the tension largest?
(b)  If the yield strength (force per area) of the cable is 10 MPa, what minimum diameter of the cable could be used?

Solution: