Problem 1 (20 points)
A 1.72-kg block, which starts with an initial speed of 0.326 m/s at the top of a ramp, slides down the ramp, which is inclined at an angle of 31.3° above the horizontal. The top of the ramp is 0.781 m higher than the bottom of the ramp.
The block reaches the bottom of the ramp with a speed of 1.23 m/s. The block then slides along a horizontal smooth surface and collides with a spring with spring constant k=174 N/m. (a) Find the coefficient of kinetic friction between the block and the ramp. (b) Find the maximum distanced that the block compresses the spring before coming to rest.
Problem 2 (20 points)
A 7.51-kg air cannon, which holds three
0.0733-kg steel balls, sits on top of a
1.22-kg cart, as shown in the figure below. Initially the air cannon and cart are at rest. The air cannon now fires the three steel balls, one at a time in rapid fire. The speed of each ball relative to the cannon is 35.2 m/s. What is the recoil speed of the air cannon and cart relative to the ground after all three steel balls are fired, assuming the cart moves without friction.
Problem 3 (20 points)
One hockey player of mass m1 = 56.9-kg
skating with speed of vii = 2.06-m/s
collides with another hockey player of
mass m2 = 75.2-kg skating at speed v2i =
5.15-m/s and they stick together. Assume hockey player m1 initially travels along the +× direction. The angle between their initial directions of travel was 132°. (a)
What is the final speed of the hockey players after the collision? (b) What is the angle of the final velocity of the hockey players with the x-axis?
Problem 4 (20 points)
The force shown in the figure below acts on a 0.73-kg object that moves along the x-axis. The object has an initial kinetic
energy 2.1-J at x = 1.5 m. (a) Suppose the
obiect moves from x = 1.5 m to x = 3.0 m.
Find the speed of the object at × = 3.0 m.
(b) Instead, suppose the object moves
from x = 1.5 m to x = O. Find the speed of
the object at × = 0. Assume the graph
increases linearly from F = O at x = O to F
= 0.80 N at × = 1.5 m; the force is
constant at 0.80 N from x = 1.5 m to x =
3.0 m; and the force decreases linearly from F = 0.80 N at × = 3.0 m to F = O at x
= 4.50 m.
Problem 5 (20 points)
A long thin rod can rotate about one of its end points. The mass of the rod 0.152-kg and its length is 0.31-m. The rod is initially rotating at 0.22 rev/s. A motor is turned on at the rod speeds up uniformly to 0.45 rev/s in 1.6 min. (a) How many revolutions did the rod make in 1.6 min?
(b) How many revolutions must the rod make for into angular speed to increase from 0.22 rev/s to 0.45 rev/s? (c) How much time does it take for the rod's angular speed to increase from 0.45 rev/s to 0.78 rev/s? (d) When the rod rotates at angular speed of 0.78 rev/s, what is the linear speed at the outer endpoint of the rod (i.e. the endpoint on the opposite side from the axis of rotation)? (e) What is the rotational kinetic energy of the rod when its angular speed is 0.78 rev/s? Express rotational kinetic energy in Joules.
Assume constant angular acceleration.
Problem 6 (20 points)
A hollow sphere with a radius of 0.18 m is initially at the top of an incline. The hollow sphere rolls without slipping down the incline with an initial linear speed of 0.23 m/s. After rolling down the ramp, the hollow sphere rolls off the edge of a horizontal table and lands on the surface of a pool of water at a point 1.1 m away from the edge of the table. Suppose the edge of the table is 0.65 m above the pool of water. (a) How long (what time interval) does it take for the sphere to go from the edge of the table to land in the water? (b)
What horizontal linear speed does the the sphere have when it reaches the bottom of the incline? (c) What is the height, h, of the incline relative to the surface of the table? (d) What is the angular speed of the sphere when it is at the bottom of the incline (right before it leaves the table)? (e)
How many revolutions does the sphere make while it flies from the edge of the table to the pool of water?