PHY 2048C - Work & Energy Examples

1. Consider the path shown below for the ball of mass 2 kg. The ball is initially placed on a spring with a compression of 1 m and let go. After it is accelerated by the spring it slides along a frictionless surface. It then encounters a patch 2 m long which has a coefficient of friction equal to 0.1. Answer the following questions for the situation shown.

a) Find the velocity of the ball at point A just after it leaves the initial spring on the left .

b) Calculate the velocity of the ball at point B after it has passes over the 2m patch of friction.

c) The ball hits the spring on the right and compresses it by 0.5 m at point C. Find the velocity of the ball at this point.

2. The potential energy of a certian particle moving in one dimension is given by

U=20x²-2x⁴.

Assume all constants reflect SI units.

a) Calculate the force on the particle as a function of position.

b) Identify any equilibrium points and the force on the particle at each equilibrium point. Are these stable or unstable equilibrium points. (If you use a graph, be sure to make a rough sketch and explain your reasoning).

c) How much work is done by the force in moving from x = 0 to x = 2.

d) What is the velocity of the particle at x=2 if it starts at the origin from rest?

3. Consider the act of sliding down a hill in a 80 kg sled. At the top of a hill, a force of 500 N is used to accelerate the sled for a distance of 10 m. The sled then slides down a hill a distance of 20m where it then encounters a patch with a coefficient of kinetic friction equal to 0.2 for a distance of 30 m. After sliding over the patch of friction, the sled hits a snow bank.

a) Calculate the velocity of the sled at point A (after it has been pushed for 10 m).

b) What is the velocity of the sled at point B (at the bottom of the hill)?

c) Find the velocity of the sled at point C after it has slid over the patch of friction.

d) If the deformation of the snow bank can be thought of as a spring with force constant k=200 N/m, then find how far within the snow bank the sled travels.

4. The potential energy of a 4 kg particle moving in the x-direction is given by

U=5-20x+1/4x⁴ (assume standard SI units).

The particle starts off at the origin at with a velocity of 3 m/s.

a) Find the force on the particle as a function of position.

b) Calculate the total mechanical energy at the origin.

c) What is the velocity of the particle when it resches x=3 m.

d) Locate any equilibrium points for the particle. Indicate whether these points are stable, unstable or neutral and explain why.

5. The potential for a particle traveling in one dimension is

U = 2x + 32/x (J)

i) If the particle starts from rest at x=1, what is the velocity of the particle when x=4.

ii) Calculate the force on the particle at any point on its path.

iii) Calculate the equilibrim points due to this potential.

6. A 10 kg ball slides down an incline starting at 2 m/s at a height of 20 m above the horizontal surface below and travels as shown below.

d=20 m

a) What is the velocity of the ball when it reaches the bottom of the incline?

b) If the spring shown in the problem has a force constant k=50,000 N/m, then what is the maximum compression of the spring if the horizontal surface is frictionless?

c) If the horizontal surface contains a rough 50 m patch which is characterized by a coefficient of friction equal to 0.2, then what is the compression of the spring in this case?