Energy Problems Quiz

n this page I put together a collection of energy problems to help you understand energy better. The required equations and background reading to solve these problems are given on the conservation of energy page and the kinetic energy page.

Problem # 1

A solid ball of radius R rolls without slipping down a series of hills, as shown. Beyond the starting point, at what peak or valley is the kinetic energy of the ball minimized? Answer: The location is at peak h3

Problem # 2

From the previous problem, find an equation for the speed of the ball at the determined location, assuming the ball starts rolling from rest.

Problem # 3

A block of mass M is traveling on a horizontal frictionless surface at a velocity V, before colliding with a spring. The spring has a stiffness k. How much is the spring compressed? Answer: V(M/k)1/2

Problem # 4

In the previous problem, the coefficient of kinetic friction is μk, between block and surface. How much is the spring compressed?

Problem # 5

A block of mass M falls onto a spring. Before colliding with the spring the velocity of the block is V. The spring has a stiffness k. How much is the spring compressed? Answer: Mg/k+(1/k)[(Mg)2+MkV2]1/2

Problem # 6

In the previous problem, 10% of the kinetic energy is lost during the collision between block and spring. How much is the spring compressed? Answer: Mg/k+(1/k)[(Mg)2+0.90×MkV2]1/2

Problem # 7

A block of mass M is traveling on a surface inclined at angle θ. The coefficient of kinetic friction is μk, between block and surface. The velocity of the block is V before colliding with a spring. The spring has a stiffness k. How much is the spring compressed?

Problem # 8

Two people are at the gym having a debate about how much energy is required to bench press a certain weight for one repetition. One person says that the energy required depends on how fast the weight is lifted. The other person says that it doesn’t matter. Using an energy analysis determine which person is correct.

The hints and answers for these energy problems will be given next.

Hints And Answers For Energy Problems

Hint and answer for Problem # 2

The kinetic energy of the ball consists of translational and rotational kinetic energy. Using conservation of energy we can solve for the speed of the ball. Therefore, mgh1 = mgh3+(1/2)m(VG)2+(1/2)IGw2, where m is the mass of the ball, VG is the velocity of the center of mass of the ball, w is the angular velocity of the ball, and IG is the rotational inertia of the ball about its center of mass. For no slipping w = (VG)/R, where R is the radius of the ball. And IG = (2/5)mR2. Combining these three equations we can solve for VG.

Hint and answer for Problem # 4

The friction force produces an additional energy term in the energy equation. The energy equation is (1/2)MV2 = (1/2)ks2+Mgμks, where Mg is the normal force, and s is the sliding distance, which is equal to the spring compression. This is a quadratic equation with two roots. We reject the negative root. The positive root is the solution.