Exercises on Conservation of Energy

Helena Dedic

Beware: Many of the solutions to these exercises use $g=10m/s^{2}$ !

Exercise 1

A "Hot Wheels" is a toy car that runs on a track as shown. Assume that the car rolls down the track without slipping and so, there are no losses of energy due to the friction. Show that if the car is released from rest, it must be at a height $h=2.5R$ above the bottom of the loop to go around the loop-the-loop without falling down at the top.

• SOLUTION EX 1
  
  

Exercise 2

A projectile of mass 0.25 kg moving at 24 m/s collides with and sticks to a 1.75-kg block that is connected to a spring for which $k=40$ N/m. The block is initially at rest but starts to slide on a rough surface immediately after the collision. If the maximum compression of the spring is 0.5 m, what is the force of friction on the block?

• SOLUTION EX 2
  
  

Exercise 3

A 3.2-kg trolley initially moving at 5 m/s at a height of 4 m encounters a hill of height 5 m, as shown below. At a later point there is a horizontal spring ($k=120$ N/m) at a height of 2 m.

(a) Does the trolley reach the spring?

(b) If so, what is the maximum compression? Ignore frictional losses and the rotational energy of the wheels.

• SOLUTION EX 3
  
  

Exercise 4

A 500-g block is dropped from a height of 60 cm above the top of a vertical spring whose stiffness constant is $k=120$ N/m. Find the maximum compression.

• SOLUTION EX 4
  
  

Exercise 5

Two blocks with masses $m_{1}=5$ kg and $m_{2}=3$ kg are connected by a light thread that passes over frictionless pegs, as in the figure below. The smaller mass is attached to a spring ($k=32$ N/m). If the system starts at rest with the spring unextended, find:

(a) the maximum displacement of the larger mass;

(b) the speed of the larger mass after it has fallen 1 m.

• SOLUTION EX 5
  
  

Exercise 6

A 75-kg parachutist is attached to an 8-kg parachute. She jumps from a plane flying at 140 km/h at an altitude of 1 km and immediately opens the parachute. If she lands vertically at 7 m/s, find the work done by the parachute on the air.

• SOLUTION EX 6