Practice Cumulative Problems
1. A pier is built on a cliff as shown below. A 35 kg ball is lowered from the pier by attaching it (through the use of a massless pulley and rope) to a 15 kg block sitting on the pier. Note that the coefficients of friction between the block and the pier are mS =0.2 and mK=0.1. In operation, the system is released from rest and the block is allowed to slide 20 m until it hits a stop piece at the end of the pier.
a) Draw free body diagrams for the masses.
b) Does the system of masses move? Clearly show your reasoning for any credit.
c) Find the acceleration of the masses and the tension in the rope.
d) If the block is released from rest, then what is the velocity of the block after it has traveled 20 m to the end of the pier?
e) If the block stops after it hits the stop-piece at the end of the pier, then find the impulse delivered to the block from the stop piece.
f) It takes 0.075 seconds for the block to be stopped at the end of the pier. Calculate the force of impact between the block and the stop-piece.
2. A 6 kg projectile is launched from the ground at an angle of 30o with the horizontal and an initial speed of 40 m/s. At the top of its flight, it explodes into two parts with masses 2 and 4 kg. Just after the explosion, both pieces have a velocity in the horizontal direction.
a) If the smaller piece recoils horizontally backward with a speed of 15 m/s just after the explosion, then what is the velocity of the larger piece just after the explosion?
c) Where is the position of the center of mass when the two pieces land?
d) The energy of the explosion is the change in energy just before and just after the explosion. Calculate the energy of the explosion.
e) If all of the energy of the explosion goes into spinning the smaller piece, then calculate the final angular velocity of this piece after the explosion. The moment of inertia of the smaller one is 10-4 kg m2.
3. A 2 kg block is released from rest at the top of a 30o inclined plane. It slides along the surface for 4 m and then encounters a massless spring with k=100 N/m which lies on a frictionless area and is fixed at the bottom of the plane.
a) Calculate the maximum compression of the spring at the bottom of the plane.
b) After compressing the spring, the block will experience a force which causes it to go in the opposite direction up the plane. Find the speed of the block as it returns to the region of friction.
c) How far up the plane did the block travel?
4. Projectile motion. Work the following projectile motion problem using methods learned from the Impulse-Momentum Theorem. Full credit will be given only if the problem is worked using the method asked for.
A 0.02 kg projectile is fired off of a 100 m cliff with an initial horizontal velocity of 120 m/s.
a) How long does it take to reach the ground below?
b) Calculate the vertical impulse delivered to the projectile from the time it is fired until the moment it hits the ground.
c) Using the impulse found above, find the velocity of the projectile just before it hits the ground.
d) Briefly explain whether the vector momentum is conserved and why.
5. You are in charge of firing a cannon at a lookout tower which is 200 m away. The cannon is tilted at 45o.
a) If you want the cannonball to hit the tower at the same height it leaves the cannon, what are the components of the velocity (vx and vy) necessary to hit your target.
b) What is the impulse delivered to the cannonball in order to reach the required velocity?
c) What average force must be applied to the cannonball in order for it to hit its target if the gunpowder takes 0.03 seconds to explode?
d) What is the force exerted on the cannon?
e) If the cannon has a mass of 100 kg, then what is the acceleration of the cannon and the normal force exerted by the ground on the cannon as the ball is fired?
6. A 200 kg Santa is falling down a 20 m chimney. The walls of the chimney provide a constant frictional force of 200 N up the chimney, opposing Santa’s motion.
a) Draw a free body diagram for Santa.
b) What is Santa’s acceleration down the chimney?
c) If he starts from rest at the top of the chimney, how fast is he going at the bottom of the chimney?
d) What was Santa’s initial mechanical energy at the top of the chimney?
e) Use the generalized work-energy theorem to figure out the non-conservative work done by friction. Partial credit will be awarded for using the definition of work instead of the generalized work-energy theorem.