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3.2.1 Introduction to Work & Energy
Definitions
• energy: __________________________________________________________
o Unit: ________________________
• work: _____________________________________________________________
o requires: ______________________
o Unit: ________________________
Equation
Example #1 – No Motion
• A student is trying to push a heavy crate across the floor, but it does not move. Is work being done ‘on the crate’?
Example #2 – Flat Ground
• A box is pushed with a force of 100 newtons across a frictionless surface for a distance of 10 meters. How much work is done on the box?
Example #3 – Incline
• A box is pulled up a 3.0 meter long plane that is inclined at 20° using a force of 10 newtons. How much work is done in moving the box?
3.0 m 10 N 100 N
Example #4 – Force on an Angle
• A sled is pulled a horizontal distance of 4.0 meters across a frictionless, horizontal surface using a force of 15 newtons at an angle of 30° above the horizontal. How much work is done on the sled?
PRACTICE
1) If 2000 joules of work are done in moving an object a distance of 500 meters, determine the average force that must be exerted on the object.
2) Calculate the amount of work done in pulling an object 4.0 meters up a 60° frictionless incline if the force directed along the plane is 30 newtons.
3) A 100 newton block is placed on a frictionless 30° incline.
a. Draw the weight and normal force vectors acting on the block.
b. Determine the force needed to move the block up the plane at a constant velocity.
c. If the block is moved 2.0 meters along the plane using this force, how much work is done on the block?
15N
4.0 m 30°
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3.2.2 Work/Energy Principle
Definitions – Types of Energy
• potential energy: __________________________________________________
o can be in a ___________________ or _____________________
o gravitational PE is stored in gravitational fields Equation
o elastic PE is stored in elastic objects (springs) Equation
Example #1
• A 30 kilogram object is positioned 2.0 meters above the surface of the Earth. What is the gravitational potential energy of the object?
Example #2
• The length of a spring with a spring constant of 100 newtons per meter is changed by 4.0 centimeters. What is the elastic potential energy stored in the spring?
• kinetic energy: _____________________________________________________
o depends on mass and velocity Equation
Example #3
• A 5.0 kilogram toy car is moving at 2.0 meters per second. What is the kinetic energy of the toy car?
• internal energy: ___________________________________________________
Variable
Work/Energy Principle
When work is done ‘on a system’ there is a net gain of one or more types of energy by that system. Energy types can change within a system at any time.
WORK W = Fd = ΔET
Gravitational PE Elastic PE
Kinetic Energy Internal Energy if ________________ changes
if ________________ changes if ________________ changes if ________________ changes
Scenario #1
• A car is moving along a flat surface with an increasing speed. What type(s) of energy are changing?
Scenario #2
• A car is moving up an incline with constant speed. What type(s) of energy are changing?
Scenario #3
• A car is applying its brakes and slowing down as it moves down a slope. What type(s) of energy are changing?
Scenario #4
• A car is moving on a flat surface toward a wall and is stopped by a spring on its front. What type(s) of energy are changing?
Scenario #5
• A rocket flies upward with an increasing velocity [ignore air resistance]. What type(s) of energy are changing?
PRACTICE
1) A 2.0 kilogram object that is initially at rest is accelerated to a speed of 5.0 meters per second.
a. What are the initial and final kinetic energies of the object?
b. How much work needed to be done on the object?
2) Determine the amount of work needed to move a 5.0 newton object from a 1.5 meter high shelf to a 4.0 meter high shelf.
3) A 3.0 kilogram object is pushed with a force of 2.0 newtons over a distance of 6.0 meters. How much kinetic energy is gained by the object?
4) Determine the change in the length of a spring with a spring constant of 50 newtons per meter when 400 joules of work are done on it.
PEg PES KE Q X = no change
↑ = increase ↓ = decrease
? = can’t be determined
PEg PES KE Q
X = no change ↑ = increase ↓ = decrease
? = can’t be determined
PEg PES KE Q
X = no change ↑ = increase ↓ = decrease
? = can’t be determined
PEg PES KE Q
X = no change ↑ = increase ↓ = decrease
? = can’t be determined
PEg PES KE Q
X = no change ↑ = increase ↓ = decrease
5 | P a g e - 75 joules of work are done in lifting a 2.5 kilogram object. What change in
height will the object experience?
[3.05m]
- What is the spring constant of a spring that requires 0.4 joules of work to be compressed a distance of 0.2 meter?
[20N/m]
- How much work is needed to change the speed of a 2.0 kilogram object from
4.0 meters per second to 10 meters per second?
[84J]
- Sketch the approproate shape for each of the graphs below.
[direct; polynomial;
h PE
v KE
x PES
Review Questions
3.2.2 – Use equations to calculate work; force; distance; PE; and/or KEin frictionless systems Identify/sketch graphs that related variables involved with energy calculations.
- How much work is done on an object if 35 newtons of force are applied to it as it is moved 10 meters?
[35J]
- What amount of kinetic energy does a 4.0 kilogram object have when moving at 3.0 meters per second?
[18J]
- A 10 newton force is applied to an object at an angle of 30° above horizontal. If the object is moved a distance of 2.0 meters horizontally along a frictionless surface using this force, what amount of kinetic energy is gained by the object?
[17J]
- A 5.0 kilogram object initially at rest is moved 4.0 meters across a frictionless surface with a force of 20 newtons.
o What amount of kinetic energy does the object gain?
[80J]
o What final speed will the object have? .
3.2.3 Conservation of Energy
Law of Conservation of Energy
The total amount of energy in a closed system must remain constant – energy is never created or destroyed.
Equation
Example #1 – Cliff Diver
• A 500 newton cliff diver climbs to a height of 50 meters above the water.
• What is his total energy at the top of his climb?
• What is his potential energy at the point where he is 40 meters above the water’s surface? What is his kinetic energy at that point?
Example #2 – Skier
• A skier begins at rest at the top of a snowy, frictionless hill. He slides down the hill and after reaching the bottom slides across a long stretch of unpacked snow.
o At which point is the total energy completely potential / kinetic / internal?
Example #2 – Carts and Hills
• A 4.0 kilogram cart operating on a frictionless track has a speed of 5.0 meters per second while at the top of a 2.0 meter high hill.
o What is the cart’s total energy at the top of the hill?
o What is the cart’s speed at the bottom of the hill? unpacked snow
Total Mechanical Energy = ___________________
v = 0 v = 0
50 m
0 m 25 m
PE
PE PE
KE
KE KE
2.0 m 5.0 m/s
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Example #3 – Spring Toy
• A spring toy with a mass of 2.0 grams has a spring with a spring constant of 120 newtons per meter. The toy is compressed 4.0 centimeters.
o How much energy can be stored in the toy’s spring?
o What maximum height will the spring toy reach if it is released?
Example #4 – Pendulum
• Label points at which potential and kinetic energies are maximum/minimum.
PRACTICE
1) An astronaut standing on the Earth throws a 0.2 kilogram wrench into the air with an initial speed of 2.0 meters per second
a. What potential energy will the wrench have when it reaches its maximum height?
b. Compare the maximum height that the wrench would reach on Earth to the maximum height that it would reach on the Moon where acceleration due to gravity is 1.62 meters per second2.
2) A 3.0 kilogram cart that is moving at 4.0 meters per second moves to the top of a 0.5 meter high hill. What kinetic energy will the cart have when it reaches the top of the hill?
h
3.2.4 Work to Overcome Friction
Energy “Loss”
Energy can never truly be destroyed, but it can be converted into non-mechanical forms (mainly heat) that are not ‘useful’
We now consider problems in which Q ≠ 0. Example #1 – Lifting a Load
In the real world it takes more energy to lift a load than the gain in potential energy would suggest.
• 5200 joules of work are done in lifting a 50 kilogram object a distance of 10 meters.
o Determine the amount of potential energy gained by lifting this object the given height.
o How much energy must have been used to overcome friction?
Example #2 – Missing KE
In the real world potential energy is not completely converted into kinetic energy when objects lose height.
• A 4.0 kilogram object begins at rest at the top of an incline that is 2.0 meters above the ground and slides down to the bottom. The object reaches a speed of 5.0 meters per second at the bottom of the hill.
o Determine the total energy of the object at the top of the incline.
o Determine the amount of energy used to overcome friction as the block slides down the incline.
Example #3 – Constant Velocity
In the real world we must constantly add energy to a system (do work) to keep things moving at a constant speed.
• A 2.0 kilogram object is pulled along a flat surface at a constant speed with a force of 30 newtons.
o Calculate the amount of work done in moving the object 4.0 meters.
o How much work must have done against friction in moving this object?
Q
KE
PE
E
T
9 | P a g e PRACTICE
1) A 50 newton cart that begins at rest at the top of a 3.0 meter high hill reaches a kinetic energy of 120 joules as it reaches the bottom of the hill. How much internal energy is generated as the cart moves down the hill?
2) A 3.0 kilogram block is dropped from a height of 10 meters. If air resistance does 200 joules of work on the block as it falls, what kinetic energy will the block have as it reaches the ground?
- A 4.0 kilogram object moving at 15 meters per second is slowed down to a speed of 6.0 meters per second by a friction force of 20 newtons.
o How much internal energy is produced during this change?
[378J]
o What distance will the object travel during this motion?
[18.9m]
- 500 joules of energy are used in lifting a 10 kilogram object a distance of 3.0 meters. What amount of work is done by friction in this system?
[205.7J]
- A 6.0 newton object is lifted a distance of 10 meters. If 100 joules of work are done by friction during this lift, what is the total amount of energy expended as the object is lifted?
[160J]
- A 2.0 kilogram object at rest at the top of a frictionless 4.0 meter high hill is allowed to slide to the bottom. What speed will the object reach at the bottom of the hill?
[6.3m/s] Review Questions
3.2.3 – Use the principle of Conservation of Energy to determine changes in KE; PE; internal energy; speed and /or height.
- How much work is needed to accelerate a 5.0 kilogram object moving at 3.0 meters per second on a frictionless surface to a speed of 10 meters per second?
- A 500 newton per meter spring is compressed a distance of 0.06 meter then used to shoot a 0.02 kilogram object along a horizontal surface. The object is launched with a speed of 6.0 meters per second.
o Determine the amount of energy stored in the spring.
[0.9J]
o Determine the kinetic energy of the object at launch.
[0.36J]
o What amount of internal energy is generated in this system?
[0.54J]
- A 5.0 kilogram object at the top of a 2.0 meter high hill gains 15 joules of kinetic energy as it slides to the bottom of the hill. How much work was done by friction as the object slid down the hill?
[83.1J]
- A spring with a spring constant of 1000 newtons per meter is used to launch a 3.0 kilogram object vertically. The spring is compressed 0.10 meter before the launch.
o What amount of energy will be stored in the spring before launch?
[5J]
o How high will the object go?
[0.17m]
- A ball slides down from rest starting at point A on the frictionless track shown below.
o At which point is gravitational potential energy a maximum? _______ o At which point is kinetic energy a maximum? ________
o Compare the ball’s speed at point C to its speed at point D. ________
[A; B; C < D]
A
B
C
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3.2.5 Mechanical Power
Definitions
• power: the rate at which…
o _______________________________________________
o _______________________________________________
o Unit: ________________________
Equation
Example #1 – Power Plant
• A power plant produces 200 mega joules of energy in 30 seconds. What is the power output of the plant?
Example #2 – Dog Sled
• A team of dogs pull a sled with an average force of 3000 newtons across a distance of 200 meters in 75 seconds. What is their power output during this time?
Example #3 – Elevator Motor
• A motor pulls an 800 newton elevator upward with an average velocity of 3.0 meters per second.
o What power must be supplied by the motor?
o How can the demand for power be reduced?
PRACTICE
1) A 50 kilogram student who is capable of generating 700 watts of power climbs a set of stairs that is 6.5 meters high.
a. Calculate the amount of work that the student needs to do in order to climb the stairs.
b. Determine the amount of time that it will take for the student to climb the stairs.
2) Compare the power output of a man lifting a 300 newton weight 2.0 meters in 2.0 seconds to the power output of a woman lifting a 100 newton weight 1.5 meters in 0.5 seconds.
Review Questions
3.2.3 – Use equations to determine power; work; time; force; and/or distance.
- What amount of power is generated in a system that uses 35 joules of energy in 10 seconds?
[3.5W]
- What is the power output of a motor that applies 75 newtons of force in lifting an object 5.0 meters in 2.5 seconds?
[150W]
- What is the power generated by a person who pulls a rope with a force of 100 newtons, lifting an object with an average speed of 3.5 meters per second?
[350W]
- A 3000 watt motor is used to drive a boat. How long will it take the motor to use 6.0 x 106 joules of energy?
