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Mechanical Energy

1. Work: Work is done on an object whenever a force makes that object move IN THE SAME DIRECTION as the force

The applied force and displacement must be in the same

direction

It is possible to have (1) displacement, (2) force, or (3) both,

but no work

Work is negative if the force opposes the motion (ex. friction)

Sample Problems:

1. Work is not being performed by the man in both pictures. Explain why. a. man at rest b. man in motion

2. A box is pushed 8 m across the floor with a force of 100 N. How much work is performed?

3. Find the work required to

a. lift an object with a mass of 3.0 x 103 kg to a height of 5.0 m. b. lower the same object to a position that is 1.0 m above the initial position.

Work done at an angle.

4. Calculate the work done by a horse that exerts an applied force of 100.0 N on a sleigh if the harness makes an angle of 30.0°with the ground, and the sleigh moves 30.0 m across a flat, level ice surface.

work power total mechanical energy

W = F x d = ma x d = ∆Ek elocityvma

t

mad

t

Fd

t

WP ET = Ek + Eg

kinetic energy potential energy conservation of energy

E mvk 1

2

2 E mghp Eki + Epi = Ekf + Epf

momentum impulse conservation of momentum p = mv∆ p = m∆v = F∆t p1 + p2 = p1’ + p2’

W = F∆d F = force, N ∆d = displacement of the object, m W = work, joules

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Practice Problems: 1. In each of the following examples, state whether or not any work is done by the force mentioned. Explain.

a) You pull a heavy sack along the ground. b) The force of gravity pulls you downwards when you fall. c) The tension in a string pulls on a stone when you whirl it around at a steady speed. d) The contact force of the bedroom floor stops you from falling into the room below.

2. A force of 835 N is needed to push a car across a lot. Two students push the car 35 m. How much work is done? (29 000J)

3. A stone of weight 10.0 N falls from the top of a 250 m high cliff. Calculate how much work is done by the force of gravity in pulling the stone to the foot of the hill. (2500 J)

4. A man of mass 70.0 kg climbs stairs of vertical height 2.5 m. Calculate the work done against the force of gravity. (1700 J)

5. Calculate the work done by a 50.0 N force at an angle of 30.0° to the horizontal if the box moves 10.0 m. (433 J)

6. An airplane passenger carries a 215 N suitcase upstairs, a displacement of 4.20 m vertically and 4.60 m horizontally. a) How much work does the passenger do? (903 J)

b) The same passenger carries the same suitcase back down the same stairs. How much work does the passenger do now? (-903 J)

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7. A rope is used to pull a metal box 15.0 m across the floor. The rope is held at an angle of 46.0° with the floor and a force of 628 N is used. How much work does the force on the rope do? (6540 J)

LAB: The StairmasterTM Lab

2. Power

Work has to do with a force causing a displacement. Sometimes the work P = W/t P = power, watts (W) is done quickly and other times the work is done quickly and other times W = work, joules (J)

the work is done slowly. Two people might do the same amount of work t = time, seconds (s) but one person might do the work in a shorter amount of time. The quantity that has to do with the rate at which a certain amount of work is done is known as the power

Common units for power are: kilowatts: 1 kW = 1 x 103 W horsepower: 1 hp = 746 W Sample Problems

1. Two physics students, Will N. Andable and Ben Pumpiniron, are in the weightlifting room. Will lifts the 100-pound barbell over his head 10 times in one minute; Ben lifts the 100-pound barbell over his head 10 times in 10 seconds. Which student does the most work? Which student delivers the most power? Explain your answers.

2. How much power does a crane develop doing 6.0 x 104 J of work in 5.00 min?

3. How much work can a 0.75 hp electric mixer do in 2.5 min?

4. A boy who can generate 5.0 x 103 W runs up a flight of stairs in 5.0 s. How high are the stairs if the boy has a mass of 62 kg?

5. An object with a mass of 110 kg is pushed up an inclined plane with a length of 8.0 m and a height of 3.0 m. The

force required to push the object along the inclined plane is 41 N.

a) Calculate the work done using the inclined plane. Calculate the work done in lifting the block vertically.

b) Does it take more or less work to move the block using the inclined plane? What about force?

elocityvma

t

mad

t

Fd

t

WP

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c) What is the advantage(s) of using an inclined plane? What is the disadvantage(s) of using an inclined plane? (Look

at work, force, distance to answer this)

d) The box is moved in 1 minute. Determine the power developed by the student moving the box along the ramp.

Practice Problems: 1. A rock climber wears a 7.50 kg knapsack while scaling a cliff. After 30.0 min, the climber is 8.2 m above the starting point.

a. How much work does the climber do on the backpack? (6.0 x 102 J)

b. If the climber weighs 645 N, how much work does she do lifting herself and the knapsack? (5900 J)

c. What is the average power developed by the climber lifting herself and the knapsack? (3.3 W)

2. An electric motor develops 65 kW of power as it lifts a loaded elevator 17.5 m in 35.0 s. How much force does the motor exert? (130 000 J)

3. Energy

Energy is the ability to do work.

Each time work is done, energy is transferred from the object doing the work to the object being worked on.

The energy of the system is increased by exactly the same amount as the work done.

Characteristics of energy: o Energy can be transferred from one object to another

whenever work is being done. o Energy has many forms which are interchangeable. o Energy can be stored and used at a later time to perform work. o Energy is always conserved.

W = ∆E

W = work done on an object, J ∆E = change in energy of the object, J

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Kinetic Energy

This is the energy of motion.

It is the result of work having been done on an object. 1. There are five types but we will refer to bulk

kinetic energy as kinetic energy.

The kinetic energy of a moving object depends on the object’s mass and speed:

Sample Problems:

1. What is the kinetic energy of a 6.0 kg curling stone sliding at 4.0 m/s?

2. What is the speed of a 5.44 kg shot-put if its kinetic energy is 68 J?

3. A 50.0 kg boy on a 20.0 kg bike starts at rest and accelerates to 5.0 m/s. How much work was done?

4. A figure skater catches his female partner, who is 50.0 kg. Her speed as she is about to fall into his arms is 2.0 m/s. She falls

20.0 cm while being caught. a. What was the kinetic energy of the female partner?

b. How much work did the male skater do when catching the female skater?

c. What force did the male skater apply to stop his female partner? Practice Problems:

1. Calculate the kinetic energy of a 750 kg compact car moving at 50.0 km/h.(72 000 J) NOTE: You won’t have to convert from km/h m/s on a test but you should know how to do a conversion for future science classes.

Ek = ½ mv2 Ek = kinetic energy, Joules m = mass, kg v = velocity, m/s

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a) How much work must be done on the car to slow it from 100.0 km/h to 50.0 km/h? (-220 000 J)

b) How much work must be done on the car to bring it from 50.0 km/h to rest? (-72 000 J)

2. A rifle can shoot a 4.20 g bullet at a speed of 965 m/s.a) Find the kinetic energy of the bullet. (1960 J))

b) What work is done on the bullet if it starts from rest? (1960 J)

c) If the work is done over a distance of 0.75 m, what is the average force on the bullet? (2600 N)

d) If the bullet comes to rest by pushing 1.5 cm into metal, what is the magnitude and direction of the average force it exerts? (130 000 N)

Gravitational Potential Energy

Work must be done when an object is lifted away from the Earth’s surface against the force of gravity.

The work done on the object is returned as work done by the object when it falls down.

The energy that is stored by a raised object is gravitational potential energy.

The force necessary to just lift a mass from position A to position B is given by Fg= mag g = acceleration due to gravity

The work done to life the mass the distance of h is Work = force x distance W = mgh Eg = magh Eg = gravitational potential energy, Joules

m = mass of the object, kg ag = acceleration due to gravity, m/s2 h = vertical distance, m

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Sample Problem: 1. A 15.0 kg textbook is sitting on a 1.20 m tall table. If the book is lifted 0.80 m above the table, how much gravitational

potential energy does it have: a. with respect to the table? b. with respect to the floor?

2. An archer pulls on a bow string with an average force of 240 N while drawing the arrow back a distance of 0.200 m. Calculate the potential energy of the bow arrow system. HINT: The work done to the bow is all being stored as potential energy.

Practice Problems:

1. A 90.0 kg rock climber first climbs 45 m upward to the top edge of a quarry, then, from the top, descends 85 m to the bottom. Find the potential energy of the climber a) at the edge and b) at the bottom, using the initial height as the reference level. (4.0 x 104 J at the edge; -35 000 J)

2. A 50.0 kg shell is shot from a cannon at Earth’s surface to a height of 4.00 x 102 m. a. What is the gravitational potential energy with respect to Earth’s surface of the Earth-shell system when the shell is at this

height? (196 000 J)

b. What is the change in potential energy of the system when the shell falls to a height of 2.00 x 102 m? (-98 000 J)

3. A person weighing 630 N climbs up a ladder to a height of 5.0 m. a. What work does the person do? (3200 J)

b. What is the increase in the gravitational potential energy of the person from the ground to this height? (3200 J)

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4. Conservation of Energy

The sum of the gravitational potential energy and kinetic energy is the total mechanical energy.

Ex. For a falling object, the Eg decreases, but the object gains

Et = Eg+ Ek Et = total mechanical energy, Joule

Eg = gravitational potential energy, Joules

Ek = kinetic energy, Joules

Law of Conservation of Energy: In any energy transfer or transformation of energy, the total amount of energy remains constant. Eti = Etf

Eki + Epi = Ekf + Epf

Sample Problems: 1. A 56 kg diver runs and dives from the edge of a 4.0 m cliff into the water below. If she is moving at 8.0 m/s the instant

she leaves the cliff, determine a. E

t relative to the water surface. b. the speed at which she enters the water.

2. A Physics student is dropped. If they reach the floor at a speed of 3.2 m/s, from what height did they fall?

3. A heavy object is dropped from a vertical height of 8.0 m. What is its speed when it hits the ground?

4. A roller coaster starts from rest at point A. What is its speed at point C if the track is frictionless? Practice Problems:

1. An engineer uses a single car to test the roller coaster track, shown in the diagram below. Assume that friction can be ignored and the roller coaster starts at rest. In each case, give a reason for your answer. a. Where is the gravitational potential energy the greatest?

b. Where is the kinetic energy the greatest?

c. Where is the speed the greatest?

d. Give a written description of what happens to the speed of the car as it rolls from A to B, b to C and so on to F.

2. Use the law of conservation of energy (assume no friction) to fill in the blanks at the various marked positions for the 1000-kg roller coaster car in the diagram. USE THREE SIG FIGS FOR ANSWERS OTHER THAN PE AND KE

a. At A PE = 100 000 J

KE = ________________

h = ________________

v = ________________

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b. At C PE = ________________ c. At F PE = ________________

KE = 40 000 J KE = ________________

h = ________________ h = ________________

v = ________________ v = ________________

3. A 300.0 kg snowmobile is traveling at 16 m/s when it comes to the edge of a small cliff. Since there is a deep fluffy snowdrift 2.5 m below the cliff, the driver doesn’t slow down but goes over the edge without changing speed. a. What is the total mechanical energy of the snowmobile when it leaves the edge of the cliff? (46 000 J)

b. How fast is the snowmobile going when it lands on the snowdrift? (17 m/s)

4. Several children, pretending that they are playing in the jungle, suspend a rope from an overhead tree limb. A child of mass

40.0 kg running at 8.0 m/s grabs the rope and swings off the level ground. a. What maximum height does the child reach? (3.3 m)

b. How fast would a 30.0 kg child have to run to reach the same height as the 40.0 kg child? (8.0 m/s)

5. An amusement park has a 10.0 m long slide for which participants are given a cloth sack to sit on. The top of the slide is 6.0 m high. Determine the speed attained at the bottom of the slide by a 30.0 kg child. Assume that the child starts at rest and make the unrealistic assumption that friction can be ignored in this case. (11 m/s)

6. A boy fires a 60.0 g pebble with his slingshot. The pebble leaves the slingshot at 35 m/s.

a. How high up will the pebble rise if it is fired straight up? (62 m)

b. If the pebble is fired so that it goes in an arc and has a speed of 10.0 m/s at its maximum height, what will the maximum height be? (57 m)

c. CHALLENGE YOURSELF : At what angle was the pebble in (b) released? HINT: What direction is the speed of 10.0 m/s

for? (73)

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http://www.cstephenmurray.com/onlinequizes/physics/workandenergy/kineticvspotentialenergy.htm

5. Momentum

The announcer says, “Going into the Labour Day Classic, the Saskatchewan Roughriders have the momentum.” What does this mean?

An object can have a large momentum if its mass or velocity is large. o Consider a Mack truck and a roller skate moving down the street at the same speed.

o Which has a greater momentum?

o If the Mack truck were at rest, which would have a the greater momentum

Both variables - mass and velocity - are important in comparing the momentum of two objects.

Sample Problems: Determine the momentum of a ...

a. 60.0-kg halfback moving eastward at 9.0 m/s. b. 1000-kg car moving northward at 20 m/s.

Force and Momentum Any object with momentum is going to be hard to stop. To stop such an object, it is necessary to apply a force against its motion for a given period of time. The more momentum that an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt. As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.

A net force must be applied to change the momentum of an object; this change is called impulse.

This equation is known as the impulse-momentum theorem.

Often, the force is not constant during the time interval, so F is an average force.

∆p is a vector – same direction as the applied force.

Applications of Impulse & Momentum 1. Increasing Momentum

e.g. golfing, baseball need a positive ∆p apply a large force “follow through” to increase time of contact

Any moving object has momentum which

depends on:

1.

2.

Where: ρ =

m =

v =

Momentum is a ___________ quantity, meaning is has both _______________ and _________________. The momentum

is in the same direction as the ____________________. The units are _____________ .

impulse = change in momentum ∆p = F∆t = m∆v = mv2 – mv1 = N●s

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2. Decreasing Momentum over a Long Time e.g. crashing into a concrete wall or a haystack, airbags, padded dashboards, landing straight-legged vs. bending legs,

“taking” a hit as a boxer or hockey player in either case, you will have the same change in momentum (impulse = Ft) by increasing the time, the force of contact decreases

3. Decreasing Momentum over a Short Time

e.g. karate expert breaking bricks contrary to #2, by decreasing the time, the force of contact increases

Sample Problems: 1. A hockey puck of mass 0.20 kg is sliding along a smooth, flat section of ice at 18 m/s when it encounters some snow. After 2.5

s of sliding through the snow, it returns to smooth ice, continuing at a speed of 10.0 m/s. a. What is the initial momentum of the puck?

b. What impulse does the snow exert on the puck?

c. What average frictional force does the snow exert on the puck?

2. An air hockey disc of mass 0.50 kg moves across an air table at a speed of 2.4 m/s when it bumps into an elastic band stretched between two fixed posts. If the elastic band exerts a force of 1.4 N on the disc for 1.5 s, what will be the final velocity of the disc?

Practice Problems: 1. A compact car, mass 725 kg, is moving at 100.0 km/h [forward].

a. Find its momentum (20 100 kg•m/s [forward]) b. At what velocity is the momentum of a larger car, mass 2175 kg, equal to that of the smaller car? (9.26 m/s [forward])

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2. A snowmobile has a mass of 250 kg. A constant force is exerted on it for 60.0 s. The snowmobile’s initial velocity is 6.00 m/s and its final velocity 28.0 m/s. a. What is its change in momentum? (5500 kg•m/s [forward]) b. What is the magnitude of the force exerted on it? (92 N [forward])

3. The brakes exert a 640 N force on a car weighing 15 680 N and moving at 20.0 m/s. The car finally stops. a. What is the car’s mass? (1598 kg) b. What is its initial momentum? (3.20 x 104 kg•m/s [forward]) c. What is the change in the car’s momentum? (-3.20 x 104 kg•m/s ; 3.20 x 104 kg•m/s [back]) d. How long does the braking force act on the car to bring it to a halt? (5.0 x 101 s)

Law of Conservation of Momentum: Predict motion following a collision using the Law of Conservation of Momentum. Collisions can be grouped into two categories, elastic and inelastic. Elastic Collisions:

• Kinetic Energy is _______________

Inelastic Collisions:

• Kinetic Energy is not conserved

• In a perfectly inelastic collision the objects

__________________ ______________________.

In reality collisions are generally somewhere in between perfectly elastic and perfectly inelastic. As a matter of fact, it is impossible for a macroscopic collision to ever be perfectly elastic. Perfectly elastic collisions can only occur at the atomic or subatomic level.

Why can’t macroscopic collisions ever be truly elastic?

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The momentum of any closed, isolated system (no external forces act on the system) does not change, so p1 = p2

p1 + p2 = p’1 + p’2 Fnet = 0 ∆ptotal = 0 Sample Problems: 1. A loaded railway car (m = 6000 kg) rolls to the right at 2.0 m/s. It collides, and couples with an empty freight car (m = 3000 kg)

that is rolling to the left on the same track at 3.0 m/s. What is the speed and direction of the pair after the collision?

2. Calculate the recoil velocity of an unconstrained rifle of mass 5.0 kg after it shoots a 50.0 g bullet at a speed of 300.0 m/s, with

respect to the earth.

Practice Problems: 1. A 0.105 kg hockey puck moving at 48 m/s is caught by a 75 kg goalie at rest. With what speed does the goalie slide on the ice

after he catches the puck? (0.067 m/s)

2. A 35.0 g bullet strikes a 5.0 kg stationary wooden block and embeds itself in the block. The block and bullet fly off together at 8.6 m/s. What was the original velocity of the bullet? (1200 m/s [forwad])

3. A 35.0 g bullet moving at 475 m/s strikes a 2.5 kg wooden block. The bullet passes through the block, leaving at 275 m/s. The block was at rest when it was hit. How fast is it moving when the bullet leaves? (2.8 m/s [forward])

4. A 0.50 kg ball traveling at 6.0 m/s collides head-on with a 1.00 kg ball moving in the opposite direction at a velocity of -12.0m/s. The 0.50 kg ball moves away at -14 m/s after the collision. Find the velocity of the second ball. (-2.0 m/s; 2.0 m/s [back])

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Mechanical Energy Review 1. State whether or not work is being done……Explain why or why not.

a) An upward force is applied to a bucket as it is carried 20 m across the yard.

b) A person climbs stairs holding a box.

c) Superman applies a force on a truck to prevent it from moving down a hill.

d) A man holds a box.

e) A person pushes a box across the floor.

2. A 21.3-kg child positions himself on an inner-tube which is suspended by a rope attached to a strong tree limb. The child and

tube is drawn back to a height of 0.33 m. The child is released and allowed to swing to and from. Assuming negligible friction,

determine the child's speed at his lowest point in the trajectory.

3. Calculate the work required lift a 2.5-kg object to a height of 6.0 meters

4. Calculate the potential energy and the kinetic energy for an object with a mass of 5.0 kg, a velocity of 4.0 m/s and a height of

2.0 m.

5. Which one of the following is an example of an object with kinetic energy not equal to zero? A) a satellite orbit D) a stationary pendulum B) a boulder resting at the bottom of a cliff E) a drum of diesel fuel on a parked truck C) a car parked at the top of a hill

6. A ball is traveling at a speed of 20.0 m/s as it approaches the bottom of a hill. Neglect the effects of friction and determine the maximum vertical height the ball ascends to.

A) 40.8 m B) 20.4 m C) 3.70 m D) 11.2 m E) 28.5 m

7. A roller-coaster car is moving at 20.0 m/s along a straight horizontal track. Predict its

speed after climbing the 15.0 m hill shown in the figure. Neglect the effects of friction. A) 10.0 m/s B) 5.00 m/s C) 17.0 m/s D) 14.0 m/s E) 7.00 m/s

8. A skier leaves the top of a slope with an initial speed of 5.0 m/s. Her speed at the bottom of the slope is 13 m/s. What is the

height of the slope?

1. 7.3 m b. 6.4 m c. 11 m d. 1.1 m e. 4.6 m

9. A rock is thrown straight up from the surface of the earth. Which one of the following statements describes the energy

transformation of the rock as it rises? Neglect air resistance.

1. The total energy of the rock increases.

2. The kinetic energy increases and the potential energy decreases.

3. Both the kinetic energy and the potential energy of the rock remain the same.

4. The kinetic energy decreases and the potential energy increases.

5. Both the potential energy and the total energy of the rock increase.

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10. An engineer is asked to design a playground slide such that the speed a child reaches at the bottom does not exceed 6.0 m/s.

Determine the maximum height that the slide can be

a. 2.9 m b. 1.8 m c. 14 m d. 3.2 m e. 4.5 m

11. What power is needed to lift a 49 kg person a vertical distance of 5.0 m in 20.0 s?

a. 12.5 W b. 210 W c. 120 W d. 25 W e. 60 W

12. A 2.0 x 102 kg roller coaster is sitting at rest 1000.0 m above the ground. The roller coaster rolls down the hill and up an incline

of 500.0 m. Ignore friction.

a) Calculate the total mechanical energy of the roller coaster at position A.

b) How much kinetic energy is gained when it reaches level ground (position B)?

c) What is its speed when it is at position C?

d) What is its speed when it is at position B?

e) The roller coaster begins to fall. What is its potential energy when it is 500.0 m above the ground? Where did the

“lost” potential energy go?

f) What is the kinetic energy of the roller coaster when it has fallen 500.0 m?

g) Would it be beneficial for roller coaster designers to use a lighter weight material in the design of a roller coaster to

increase the speed and make for a better ride? Explain.

13. A student applies a force of 10.8 N parallel to an inclined plane to pull a 3.00-kg cart up an inclined plane at a constant speed

during a physics lab. The incline is 2.63 m long and the vertical distance the box moves is 1.5 m.

a) The student is able to pull the cart up the inclined plane in 2.5 s. Determine the power developed by the mover

pushing the box along the ramp?

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b) Calculate the work done in lifting the box vertically.

c) What is/are the advantages and disadvantages of using an inclined plane to move an object?

14. A 15.0 kg medicine ball is thrown at a velocity of 20.0 m/s to a 60.0 kg person who is at rest on ice. The person catches the

ball and subsequently slides with the ball across the ice. Determine the velocity of the person and the ball after the collision.

a. Create a before and after chart to organize your information.

Before After

Ball

Person

b. Calculate the momentum of the ball before the collision. e. Calculate the speed and direction of the pair after the collision.

c. What is the impulse of the ball? f.What can the person do to reduce the force of the impact? Explain. HINT: Look at the equation

for force to help you answer this question.

d. Suppose the collision occurs in 2.0 x 10-2 s. What is the average force exerted on the ball?

15. A 3000 kg truck moving with a velocity of 10 m/s hits a 1000 kg parked car. The impact causes the 1000 kg car to be set in motion

at 15 m/s. 1. Create a before and after chart to organize your information.

2. Assuming that momentum is conserved during the collision, determine the velocity of the truck immediately after the collision.

Before After

Truck

Car