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Conservation of Energy


Conservation of Energy: Within an isolated system, one type of energy can be transformed into another type of energy, but the total of all energies in the system is constant. The energy of a system changes by the work done on or by the system and the heat that enters or leaves the system.

Any group of objects can be defined to be a system. An isolated system is a system where neither matter nor energy can leave or enter. An external force is one that is exerted by an object outside (not included in) the system. Internal forces are forces exerted by objects within the system on objects within the system.

Etotal = (0.5)mv2 + mgh + (0.5)kx2 is unchanging if no work is done on or by the system, and no heat is gained or lost.

For example: A 10.0 g pebble is placed in a sling shot with a spring constant of 200.0 N/m and is stretched back 0.500 m. What is the maximum velocity the pebble will acquire? If shot straight up, what is the maximum height (above release) the pebble will reach?

In each situation energy is conserved if heat effects and air resistance are negligible.
At the beginning, when the slingshot is "cocked"
Etotal = (0.5)mv2 + mgh + (0.5)kx2
Etotal = (0.5)m(0)2 + mg(0) + (0.5)(200.0)(0.500)2
Etotal = 25 J

The maximum velocity is obtained after the pebble is released but before it has achieved any altitude.
Etotal = (0.5)mv2 + mgh + (0.5)kx2
25 J = (0.5)(0.0100 kg)(v)2 + mg(0) + (0.5)k(0)2
v = 70.7 m/s

At its maximum height, the velocity of the pebble is zero.
Etotal = (0.5)mv2 + mgh + (0.5)kx2
25 J = (0.5)m(0)2 + (0.010 kg)(9.8 N/kg)h + (0.5)k(0)2
h = 255 m


A. Kinetic Energy / Gravitational Potential Energy Problems

(See bottom of page for answers.)

Conservation of energy is illustrated when a cart moves up a track to the maximum height dictated by the conversion of kinetic energy into gravitational potential energy.1. A cart moving along a track 1.00 m above the floor at 3 m/s eventually reaches a higher plateau What is the maximum height of the plateau above the floor?

2. A 10.5 g bullet strikes a pendulum that consists of a block of wood of mass 3.00 kg suspended by a cord. The bullet gets embedded in the block. How fast was the bullet traveling just before impact to raise the block by 0.220 m?

3. Sheila, running 5.3m/s, grabs a vine hanging vertically from a tall tree.
a. How high can she swing upward?
b. Does the length of vine affect the answer?

4. A roller coaster at the top of a 39.0 m high vertical loop is traveling 13.8 m/s. Find the maximum speed of the cars as they move through the bottom of the loop.

5. Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction; and (b) taking friction into account. Explain why a grandfather clock has to be wound up.

6. A ball is attached to a horizontal cord of length L whose other end is fixed.
a. If the ball is released, what will be its speed at the lowest point of its path?
b. A peg is located a distance h directly below the point of attachment of the cord. If h= 0.080L, what will be the speed of the ball when it reaches the top of the circular path about the peg?

7. A projectile is fired at an upward angle of 45.0 degree from the top of a 265 m cliff with a speed of 185 m/s. What will be its maximum speed of impact with the ground below?

8. If a projectile is launched from Earth with a speed equal to the escape speed, how high above the Earth's surface is it when its speed is one third the escape speed?

9. Two pieces of space debris, each with a mass of 0.116 kg, are separated by a distance of 380 m. If re released from rest, what speed do they have when their separation has decreased to 171 m? Ignore the gravitational effects from any other objects.

10. A baseball is thrown first with an initial upward velocity of + 4.0 m/s. Later, it is thrown from the same height but with and initial downward velocity of -3.0m/s. How do the impact velocities of the baseball with the ground differ? What is its acceleration in each case?

11. A 200 g ball is thrown upwards with an initial kinetic energy of 10 Joules. What maximum height will the ball attain?

12. Bruce grasps the end of a 20.0 m long rope attached to a tree and swings. If the rope starts at an angle 35 degrees with the vertical, what is Bruce's speed at the bottom of the swing?

13. Jack and Jill, whose total mass is 120 kg, sit on a swing at the end of a 5m long rope. Initially the rope attached to their swing makes an angle of 36 degrees with the horizontal.  At the bottom of the arc, Jill, whose mass is 52 kg, steps off.  What is the maximum height Jack can reach as the swing continues?

14. A 1.9-kg block slides down a curved, frictionless ramp. The top of the ramp is 1.5 m above ground; the bottom of the ramp is 0.25 m above the ground. The block leaves the bottom of the ramp moving horizontally.
a. What horizontal distance away from the base of the ramp does it land?
b. Suppose friction on the ramp does -9.7 J of work on the block. What horizontal distance away from the base of the ramp does it land?

15. A ball bounces upward from the ground with a speed of 16 m/s and hits a wall with a speed of 12 m/s How high above the ground does the ball hit the wall? Ignore air resistance.

16. From what height would a car have to be dropped to have the same kinetic energy that it has when being driven at 100 km/h?

17. A 135 m long ramp is to be built for a ski jump. If a skier starting from rest at the top is to have a speed no faster than 19m/s at the bottom, what should be the maximum angle of inclination?

18. Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

19. Two objects, m1 = 4.50kg and m2 = 3.00kg, are connected by a light string passing over a light frictionless pulley. The object of the mass 4.50kg is released from rest 4.50m above the ground. Using the principle of conservation of energy, determine the speed of the 3.00kg object just as the 4.50kg object hits the ground.

(a) What is the escape speed on a spherical asteroid whose radius is 525km and whose gravitational acceleration at the surface is 2.7m/s2?
(b.)How far from the surface will a particle go if it leaves the asteroid's surface with a vertical speed of 1000m/s?

21. In a looping the loop setup, an object of mass m is released from rest from A with the initial height h and loops the loop in the circular track of radius R.
a) Write an expression for the initial mechanical energy at A in terms of m, g, h
b) Write an expression for energy at point B at the top of the vertical circle in terms of mass m, velocity v, radius R and g
c) If in the absence of friction the object just manages to loop the loop without losing contact with the track, what is the minimum height h from which you will need to release the object? Write the expression in terms of R .

22. A 75 kg parcel falls out of a window to a sidewalk 1 m below.
a. With what speed does it impact the pavement?
b. If the packaging provides 0.50 cm of cushioning, calculate the average force exerted on the parcel by the ground in this situation.

23. Roy was transporting balls in the trunk of a car to a clubhouse. Two boxes on the floor of the trunk each contained an equal number of balls. The balls were identical except that all the balls in one box were dimpled, while all the balls in the other box were smooth. Upon arriving, Roy realized there were no lids on the boxes, and found balls all over the trunk of the car. He observed that more dimpled balls escaped than smooth balls. Why would more dimpled balls escape than smooth balls? There was nothing else in the trunk.


B. Elastic Potential Energy Problems

(See bottom of page for answers.)

Conservation of energy is illustrated by spring elastic potential energy transforming into the kinetic energy of a frictionle

1. A 0.150 kg puck on a frictionless surface is pressed against a spring with a spring constant of 2.00 N/m. The pressure causes the spring to compress 0.30 m. What velocity will the puck attain when released?

2. A 200 g mass is attached to a spring of spring constant k. The spring is compressed 20 cm from its equilibrium value. When released the mass reaches a speed of 5 m/s. What is the spring constant (in N/m)?

3. A 1200-kg car rolling on a horizontal surface has a speed v= 65km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?

4. A 0.444 kg mass is attached to a spring with a force constant of 26.8 N/m on a frictionless horizontal surface and released from rest a distance of 3.15 cm from the equilibrium position of the spring. What is the speed of the mass when it is halfway to the equilibrium position?

5. A 34-g bullet traveling at 120m/s embeds itself in a wooden block on a smooth surface. The block then slides toward a spring and collides with it. The block compresses the spring (k=99 N/m) a maximum of 1.2 cm. Calculate the mass of the block of wood.

6. An archer is taking aim at a target that is exactly 20 m away in the horizontal direction. The target is at the same height above the ground that the arrow starts from. The mass of the arrow is 132 g. If the spring constant for the bowstring is 1500 N/m and the string is stretched 82 cm from its equilibrium, find the angle above horizontal that the arrow must be fired at to hit the target.

7. A cannon of m1=800kg (when unloaded) initially resting on a frictionless surface is loaded with a "shot" of mass m2=10kg. The cannon is aimed at mass m3=7990kg, which is connected to a massless spring of force constant k=4500N/m. The cannon is fired and the shot inelastically collides with Mass m3 and sticks in it. The combined system compresses the spring a maximum distance of d=.500m. a) Determine the speed of m2 just before it collides with m3 (You may assume that m2 travels in a straight line. b) Determine the recoil speed of the cannon.

8. A rail car of mass 4000 kg rolls downhill on tracks and on to a level section of tracks 8 m in elevation lower than the starting point. At the end of the tracks is a spring with spring constant 600000 N/m. Ignoring friction, what is the maximum compression of the spring in stopping the car?


C. Multiple Conversion Problems

(See bottom of page for answers.)

1. You throw a 0.40 kg stone straight up into the air. Initially at rest, your hand exerts a force of 150 N upwards on the stone through a distance of 0.75 m. Use the concepts of work and conservation of energy to answer the following:
a. How much work did your hand do on the stone?
b. What was the net work done on the stone?
c. What was the kinetic energy of the stone at the point of release?
d. How high above the point of release did the stone rise?
e. What was the momentum of the stone at the point of release?
f. What was the net impulse exerted on the stone?
g. How long was the net force applied to the stone during throwing?

2. Assuming negligible friction, what spring constant would be needed by the spring in a toy rifle to fire a 10 gram pellet to a height of 100 meters if the spring is initially compressed 10 cm? What is the muzzle velocity of the pellet?

Conservation of energy does not apply to the object in the diagram dragged along a rough surface.

3. Consider the system shown. The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of kinetic friction between the 8.00 kg block and the tabletop is µk = 0.22. The blocks are released from rest. Use energy methods to calculate the speed of the 6.00 kg block after it has dropped 3.50 m.

4. A heavy-duty stapling gun uses a 0.139-kg metal rod that rams against the staple to eject it. The rod is pushed by a stiff ram spring (k = 32747 N/m). The mass of this spring may be ignored. Squeezing the handle of the gun first compresses the ram spring by 3.8*10-2 m from its unstrained length and then releases it. Assuming that the ram spring is oriented vertically and is still compressed by 1.1*10-2 m when the downward-moving ram hits the staple, find the speed of the ram at the instant of contact.

5. A high diver of mass 70.0 kg jumps off a board 10.0 m above the water. If her downward motion is stopped 2.00 s after she enters the water, what average upward force did the water exert on the diver?

6. Which of the following physical quantities can be conserved (select all correct choices):
a. Energy: b. Gravity; c. Time; d. Velocity; e. Linear Momentum; f. Force; g. Angular Momentum; h. Friction; i. Total Energy of the Universe; j. Work


D. Heat / Friction Problems

(See bottom of page for answers.)

1. How much heat is generated when the brakes are used to bring a 1000-kg car to rest from a speed of 100 km/h?

2. A 0.2 kg block rests on a level surface and is attached to a horizontal spring with a spring constant of 40 N/m. The spring is initially compressed 4 cm, then released to set up simple harmonic motion. A frictional force of 0.2 N exists between the block and the surface. What is the speed of the block when it passes through the equilibrium position the first time after being released?



Selected solutions are printed below.
For solutions to all the problems on this page click here.

The gravitational potential energy of a pendulum at any point is determined by the pendulum's length and the cosine of the angle of the pendulum from vertical.

A13. The swinging couple start from a height of 5 - 5cos54° = 2.06 m
Since energy is conserved, the maximum height Jack can return to is 2.06 m.
Note: mgh = (1/2)mv2
so gh = (1/2)v2 ie. h is independent of mass.

For solutions to all the problems on this page click here.

a. The kinetic energy of (m3 + m2) is converted into elastic potential energy:
(1/2)(m3 + m2)v2 = (1/2)kx2
(1/2)(7990 + 10 )v2 = (1/2)(4500)(0.500)2
v = 0.375 m/s
Applying conservation of momentum to the collision between the shot and m3
m3v3 + m2v2 = (m3 + m2) v
(7990)(0) + (10) v2 = (7990 + 10 ) (0.375)
v2 = 300 m/s
b. Applying conservation of momentum to the explosion between the m1 and m2
(m1 + m2) v' = m1v1 + m2v2
(800)(0) = (790) v1 + (10)(300)
v1 = -3.80 m/s

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D1. How many kilocalories of heat are generated when the brakes are used to bring a 1000-kg car to rest from a speed of 100 km/h?
(100 km/h)*(1000 m/km)*(1 h / 3600 s) = 27.8 m/s
Kinetic energy of the car = (1/2)(1000 kg)(27.8 m/s)2 = 3.86*105 J
This is the energy converted to heat as the car comes to a stop.
( 3.86*105 J)(1 kcal / 4.186*103 J) = 92.2 kcal

For solutions to all the problems on this page click here.

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