25
Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work-energy principle 8.1 Conservative forces 8.2 Potential Energy

Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

Embed Size (px)

Citation preview

Page 1: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

Chapter 8

Conservation of Energy7.3 work done by a varying force7.4 kinetic Energy and work-energy principle8.1 Conservative forces8.2 Potential Energy

Page 2: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

Work done by a constant force

Units of work: Nm or Joules (J)

F

r

Work is an energy transfer that occurs when a force acts on an object that moves.

•Work is done only when force is exerted over a distance.

(no displacement=no work)

http://i.telegraph.co.uk/telegraph/multimedia/archive/01435/bmw_1435680c.jpg

•Only the part of the force parallel to the displacement does work.

Fcos

Page 3: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

Work can be positive, negative or zero

cosrFWork F

r

Fpush

x

mg

N

Fy

Fx

f

Work done byWorkGravity =

WorkNormal =

WorkFriction =

WorkFPUSH=

0

0

-fxNegative since f is opposite x

Fxcos= FxxPositive since Fx same direction as x

Work done sliding a box across a floor

Page 4: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up.

Page 5: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

In the limit that the pieces become infinitesimally narrow, the work is the area under the curve:

Or:

Page 6: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

Work done by a spring force:

You are exerting a force Fp= kx

K is the spring constant or stifness.

The force exerted by a spring is given by: called restoring force

This is called Hooke’s law

Page 7: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

Plot of F vs. x. Work done is equal to the shaded area.

Page 8: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

Example 7-5: Work done on a spring.

(a) A person pulls on a spring, stretching it 3.0 cm, which requires a maximum force of 75 N. How much work does the person do? (b) If, instead, the person compresses the spring 3.0 cm, how much work does the person do?

Page 9: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-3 Work Done by a Varying Force

Example 7-6: Force as a function of x.

where F0 = 2.0 N, x0 = 0.0070 m, and x is the position of the end of the arm. If the arm moves from x1 = 0.010 m to x2 = 0.050 m, how much work did the motor do?

A robot arm that controls the position of a video camera in an automated surveillance system is manipulated by a motor that exerts a force on the arm. The force is given by

Page 10: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

Page 11: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

If we write the acceleration in terms of the velocity and the distance, we find that the work done here is

We define the kinetic energy as:

Page 12: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

This means that the work done is equal to the change in the kinetic energy:

•This is the Work-Energy Principle

• If the net work is positive, the kinetic energy increases.

• If the net work is negative, the kinetic energy decreases.

Page 13: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules. Energy can be considered as the ability to do work:

Page 14: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

Example 7-7: Kinetic energy and work done on a baseball.

A 145-g baseball is thrown so that it acquires a speed of 25 m/s. (a) What is its kinetic energy? (b) What was the net work done on the ball to make it reach this speed, if it started from rest?

Page 15: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

Example 7-8: Work on a car, to increase its kinetic energy.

How much net work is required to accelerate a 1000-kg car from 20 m/s to 30 m/s?

The net work is the increase in kinetic energy, 2.5 x 105 J.

Page 16: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

7-4 Kinetic Energy and the Work-Energy Principle

Example 7-10: A compressed spring.

A horizontal spring has spring constant k = 360 N/m. (a) How much work is required to compress it from its uncompressed length (x = 0) to x = 11.0 cm? (b) If a 1.85-kg block is placed against the spring and the spring is released, what will be the speed of the block when it separates from the spring at x = 0? Ignore friction. (c) Repeat part (b) but assume that the block is moving on a table and that some kind of constant drag force FD = 7.0 N is acting to slow it down, such as friction (or perhaps your finger).

Page 17: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-1 Conservative and Nonconservative Forces

A force is conservative if:the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken.Example: gravity.

W=-mg (y2-y1)

Page 18: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-1 Conservative and Nonconservative Forces

Another definition of a conservative force:

a force is conservative if the net work done by the force on an object moving around any closed path is zero.

(a) (b)

Page 19: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-1 Conservative and Nonconservative Forces

If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

W = FPd

Page 20: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-1 Conservative and Nonconservative Forces

Page 21: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-2 Potential Energy

An object can have potential energy by virtue of its surroundings. Potential energy can only be defined for conservative forces

Familiar examples of potential energy:

• A wound-up spring

• A stretched elastic band

• An object at some height above the ground

Page 22: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-2 Potential Energy

In raising a mass m to a height h, the work done by the external force is

We therefore define the gravitational potential energy at a height y above some reference point:

.

.

Page 23: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-2 Potential Energy

This potential energy can become kinetic energy if the object is dropped.

Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).

If Ugrav = mgy, where do we measure y from?

It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.

Page 24: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-2 Potential EnergyExample 8-1: Potential energy changes for a roller coaster.

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1. (b) What is the change in potential energy when the car goes from point 2 to point 3? (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3.

Page 25: Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work- energy principle 8.1 Conservative forces 8.2 Potential Energy

8-2 Potential Energy

General definition of gravitational potential energy:

For any conservative force: