Energy

Energy and Work

Part 1: Energy

Reading Quiz: Question 1

1. money.

2. heat.

3. a liquid.

4. work.

5. momentum.

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9%

1%

43%

14%

Energy is a physical quantity with properties somewhat similar to


1. Kinetic energy

2. Gravitational potential energy

3. Elastic potential energy.

4. Mechanical energy.

5. Electric potential energy.

1%

1%

2%

5%

90%

Which of the following types of energies are not discussed in chapter 10, sections 10.1 10.5?


1. gravity.

2. a spring.

3. collisions.

4. tension.

5. none of the above.

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96%

2%

1%

1%

Hookes law describes the force of

Kinetic Energy and Gravitational Potential Energy

Consider an object falling in free fall:

Kinematics give us this:

Rearrange a bit:

Multiply by m:

Kinetic and Potential Energy

The kinetic energy of an object is

The gravitational potential energy of an object is

For the falling ball,

Energy Bar Charts

We can use an energy bar chart shows us how energy is transferred between kinetic and potential:

Conservation of Mechanical Energy

The sum of the kinetic and potential energy of a system is called the mechanical energy:

In the situations weve seen so far, a particles mechanical energy never changes:

This is the law of conservation of mechanical energy. Is it always true? No

ConcepTest 1

1. One-half

2. The same amount as

3. Twice

4. Four times

3%

5%

85%

7%

Two balls, one twice as heavy as the other, are dropped from the roof of the building. Just before hitting the ground, the heavier ball has

the kinetic energy of the lighter ball.

ConcepTest 2

1. 2h

2. h

3. h/2

4. Greater than h, but I cant predict an exact value

5. Less than h, but I cant predict an exact value

9%

55%

17%

12%

7%

A block is shot up a frictionless 40 slope with initial velocity v. It reaches a height h before sliding back down. The same block is shot with the same velocity up a frictionless 20 slope. On this slope, the block reaches height

ConcepTest 3

1. 1.0 m/s

2. 2.6 m/s

3. 4.4 m/s

4. 8.8 m/s

1%

7%

92%

0%

A block of mass 2.0 kg slides down a ramp from a height of 1.0 m. What is its speed at the bottom of the ramp?

ConcepTest 4

1. 1.41v

2. 2v

3. 3v

4. 4v

5. 6v

35%

35%

9%

10%

11%

A block slides down a frictionless ramp of height h. It reaches velocity v at the bottom. What is its velocity at the bottom if it slides down a ramp of height 2h?

Example Potential energy

1. -9.8 J

2. 0 J

3. 9.8 J

17%

3%

81%

Consider the 1.0 kg rock in the figure below.

What does Amber say the potential energy of the rock is?


1. -9.8 J

2. 0 J

3. 9.8 J

17%

72%

11%

Consider the 1.0 kg rock in the figure below.

What does Bill say the potential energy of the rock is?


1. -9.8 J

2. 0 J

3. 9.8 J

4%

95%

1%

The rock drops to the ground.

What does Amber say the potential energy of the rock is now?


1. -9.8 J

2. 0 J

3. 9.8 J

84%

3%

14%

The rock drops to the ground.

What does Bill say the potential energy of the rock is now?


What is the change in potential energy from both their viewpoints?


What is the speed of the rock right before it hits the ground?

Problem Solving Strategy

Example: The Ballistic Pendulum

A 10 g bullet is fired into a 1200 g wood block hanging from a 150 cm long string. The bullet embeds itself into the block, and the block then swings out to an angle of 40. What was the speed of the bullet?

Restoring Forces

Systems that exhibit restoring forces are called elastic.

A great example is a spring.

Hookes Law

Hookes law tells us the restoring force for a spring:

where s = s se is the displacement of the end of the spring from equilibrium.

Quick Example: Spring on the wall

A 20 cm long spring is attached to the wall. When pulled horizontally with a force of 100 N, the spring stretched to a length of 22 cm. What is the value of the spring constant?

Quick Example: Spring on the ceiling

The same spring is now suspended from hook and a 10.2 kg block is attached at the bottom end. How long is the stretched spring?

Quick Example: Spring on the ceiling

Elastic Potential Energy

Stretched or compressed springs have some kind of stored energy that can be transferred to kinetic energy.

Called elastic potential energy.

See the textbook for derivation:

Mechanical Energy Again

With springs and gravity, a particles mechanical energy will be

There are now two kinds of stored energy inside the system.

But the mechanical energy can still be conserved.

Example: Dropping a block onto a spring

A 20 cm tall spring with spring constant 5000 N/m is placed vertically on the ground. A 10.2 kg block is held 15 cm above the spring. The block is dropped, hits the spring, and compresses it. What is the height of the spring at the point of maximum compression?

Energy Diagrams

Potential energy is a function of position.

So we can plot the potential versus position:

Energy Diagrams

Molecular Bonds

Check out this energy diagram for an HCl molecule:

Elastic Collisions

A perfectly elastic collision is one in which the mechanical energy is conserved.

Of course, momentum will be conserved, too!

Elastic Collisions

How do we analyze this kind of collision?

With our conservation laws:

We can rearrange these for the final velocities (see text):

Three Cases of Interest

1. Billiard balls (m1 = m2)

2. Bowling ball hitting a ping-pong ball (m1 >> m2)

3. Ping pong ball hitting a bowling ball (m1

ConcepTest 5

1. greater

2. less

3. the same

3%

60%

37%

Two cars of equal mass are travelling in opposite directions. Just before colliding, the speed of one is twice that of the other. They collide and lock together, moving off as one unit. Compared to the kinetic energy of the cars before the collision, their kinetic energy after the collision is

ConcepTest 6

1. The kinetic energy of the ball is the same.

2. The momentum of the ball is the same.

3. Both the kinetic energy and the momentum of the ball are the same.

4. Neither the kinetic energy nor the momentum of the ball are the same.

5. The collision is inelastic.

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5%

60%

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A ball is thrown against a wall; it bounces off and returns with speed equal to its initial speed before striking the wall. Which of the following statements is true from before to after the collision between the ball and the wall?

ConcepTest 7

9H 7

5 1

Matched Acceptable Value: NA, Within Range: 1

Acceptable Value: NA, Acceptable Range: {1, 100}

A basketball with a tennis ball on top is dropped from a height h. How high will the tennis ball go after the basketball bounces off the ground (give your answer as a multiple of h, e.g., 2h)?

Part 2: Work

The Dot Product of Two Vectors

How do you multiply two vectors together? Like force and displacement.

One way is called the scalar or dot product:

ConcepTest 8

1. Positive

2. Negative

3. Zero

98%

1%

1%

What is the dot product of these two vectors?

A

B

ConcepTest 9

1. Positive

2. Negative

3. Zero

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0%

A

B


ConcepTest 10

1. Positive

2. Negative

3. Zero

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4%

94%

A

B


Example: Dot product 1

Compute the dot product of the two vectors.

The Dot Product of Unit Vectors

What is ...

The Dot Product of Two Vectors

We can also write the dot product in terms of vector components:

Example: Dot product 2

Compute the dot product of the two vectors using their components.

Work

A force acting on a particle does work:

ConcepTest 11

1. The 10 N force.

2. The 8 N force.

3. The 6 N force.

4. They all do the same amount of work.

4%

7%

88%

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Which force does the most work?

Example: Pulling a suitcase

A rope inclined upward at a 45 angle pulls a suitcase through the airport. The tension in the rope is 20 N. How much work does the tension do if the suitcase is pulled 100 m?

Thermal Energy

Lets compare the microphysics and macrophysics of a 500 g iron ball moving at 20 m/s.

What is the macroscopic kinetic energy of the ball?

Thermal Energy

Each iron atom inside the ball has a mass of around 9 10-26 kg and probably moves around 500 m/s.

What is the kinetic energy of each iron atom?

But theres around 5 1024 atoms inside the ball ... so thats a microscopic kinetic energy of 60,000 J!

This energy is called the thermal energy.

Its related to the temperature of the object.

Dissipative Forces

So what happens to the kinetic energy when a block slides across a surface and slows down?

Its transformed into thermal energy for both objects (block and surface).

We get an increased temperature of both objects.

Forces that result in a decrease of the macroscopic kinetic energy are called dissipative forces.

Thermal Energy and Friction

The thermal energy increase due to friction can be found from:

Note that friction and other dissipative forces always increase the thermal energy they never decrease it.

Conservation of Energy

We can combine the ideas of work and thermal energy with mechanical energy into one big law of physics:

Example: Pushing a block

1. Zero

2. Positive

3. Negative

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0%

Consider pushing a block across a rough table at a constant speed.

What is K?


1. Zero

2. Positive

3. Negative

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What is U?


1. Zero

2. Positive

3. Negative

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0%

0%


What is Eth?


1. Zero

2. Positive

3. Negative

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0%

0%


What is Wext?

So what energy transformations take place? What about energy transfers?

Example: Stretching a spring

The 5.0 kg box is attached to one end of a spring with spring constant 80 N/m. The other end of the spring is anchored to a wall. Initially the box is at rest at the springs equilibrium position. A rope with a constant tension of 100 N then pulls the box away from the wall. What is the speed of the box after it has moved 50 cm? The coefficient of friction between the box and the floor is 0.30.


1. Zero

2. Positive

3. Negative

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0%

0%

What is K?


1. Zero

2. Positive

3. Negative

0%

0%

0%

What is U?


1. Zero

2. Positive

3. Negative

0%

0%

0%

What is Eth?


1. Zero

2. Positive

3. Negative

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0%

0%

What is Wext?


Now answer the actual question

Final Example

A 5.0 kg box slides from rest down a 5.0 m high frictionless hill, across a 2.0 m wide horizontal surface, then hits a horizontal spring with spring constant 500 N/m. The other end of the spring is anchored against a wall. The ground under the spring is frictionless, but the 2.0 m wide horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.25.

How many complete trips will the box make across the rough patch before stopping?

Energy

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