chapter 5 work, energy and power

CHAPTER 5 WORK, ENERGY AND POWER prepared by Yew Sze Ling@Fiona, KML

1

Chapter 5 Work, Energy and Power

Curriculum Specification Remarks

Before After Revision

51 Work

a) Define work done by a constant force, sFW

.

(C1, C2)

b) Apply work done by a constant force and from a force-

displacement graph. (C3, C4)

5.2 Energy and Conservation of Energy

a) State the principle of conservation of energy. (C1, C2)

b) Apply the principle of conservation of energy

(mechanical energy and heat energy due to friction).

(C3, C4)

c) State and apply the work-energy theorem, KW .

5.3 Power

a) Define and use average power, t

WPav

and

instantaneous power, vFP

. (C1, C2)

b) Verify the law of conservation of energy.

(C1, C2, C3, C4)


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5.1 Work

The work done on an object by a constant force is defined to be the product of the magnitude of the displacement times the component of the force parallel to the displacement.

sFW cos

Alternatively, work done is defined as the scalar product between force and displacement of a body.

sFW

Work done is a scalar quantity.

The SI unit of work done is N m or Joule (J).

Work done is positive: if work is done on the object, i.e. energy is transferred to the object. (0° < θ < 90°)

Work done is negative: if work is done by the object, i.e. energy is transferred from the object.

( 90° < θ < 180°)

If more than one force acts on the object:

Force acts on the object Work done by specific force

Tension, T

Normal force, N

Frictional force, f

Weight, W

External force, F

Total work done,FWfNTTotal WWWWWW OR cossFW netTotal


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Example 1

A weightlifter does no work on a barbell as long as he holds it stationary.

Example 2

This weightlifter’s hands do negative work on a barbell as the barbell does positive work on his hand.

The barbell does positive work on the hands The weightlifter hand’s do negative

(work is done on the hand) work on the barbell (work is done by the

hand)


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F-s Graph

If the force acting on an object is constant, the work done by that force can be calculated by

using equation above. But quite often, situations arise in which the force is not constant but

changes with the displacement of the object. For example, the force exerted by spring varies

with displacement, in which the force needed increases with the amount of stretch or compress.

The work done by a varying force can be determined graphically.

graph under the area dsFWf

i

s

s x

5.2 Energy and Conservation of Energy

Energy can be defined as “the ability to do work”. This simple definition is not precise, nor is it

really valid for all types of energy. However, it works for mechanical energy which we discuss in this sub topic.

Energy is a scalar quantity.

The SI unit of energy is Joule (J).

There are two types of mechanical energy: potential energy and kinetic energy.

POTENTIAL ENERGY

Definition: Energy stored in a body or system because of its position, shape and state.

TRANSLATIONAL

KINETIC ENERGY

Gravitational potential energy Elastic potential energy

mghU

Definition:

Energy stored in a body or system because of its position.

2

2

1kxU S

Definition

Energy stored in elastic

materials as the result of their stretching or compressing.

2

2

1mvK

Definition

Energy of a body due to its

motion.

sFWf

i

s

s

x 0s

lim


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The Principle of Conservation of Energy

The total energy is neither increased nor decreased in any process. Energy can be transformed

from one form to another, and transferred from one object to another, but the total amount remains constant.

if EE

Additional Knowledge: Conservative and non-conservative force

Forces such as gravitational force, for which the work done does not depend on the path taken

but only on the initial and final positions, are called conservative forces. For example, it takes

the same work (mgh) to lift an object of mass m vertically to a height h as to carry it up an incline of the same vertical height.

Forces such as friction and a push or pull exerted by a person, are non-conservative forces

(or dissipative force) since any work they do depends on the path. For example, if you push a

crate across a floor from one point to another, the work you do depends on whether the path

taken is straight or is curved. Greater amounts of work are done over longer paths between the points, so that the work depends on the choice of path.

Conservation of Mechanical Energy

The total mechanical energy (E = KE+PE) of an object remains constant as the object moves,

provided that the net work done by external non-conservative forces is zero.

2222

2

1

2

1

2

1

2

1iiifff mvkxmghmvkxmgh

Example

PE KE


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Conservation of Mechanical Energy with Non-conservative (Dissipative) Force

In most of the natural processes, the mechanical energy (sum of the kinetic and potential

energies) does not remain constant but decreases. Because frictional forces reduce the

mechanical energy (but not the total energy), they are called dissipative forces.

Imagine that a book has been accelerated by your hand and is now sliding to the right on the

surface of a heavy table and slowing down due to the friction force (Figure below). Suppose the

surface is the system. Then the friction force from the sliding book does work on the surface.

However, there is no increase in the surface’s kinetic energy or the potential energy of any

system. So where is the energy?

The surface will be warmer after the book slides over it. The work that was done on the surface

has gone into warming the surface rather than increasing its speed or changing the configuration

of a system. If heat is considered as a transfer of energy, then the total energy is conserved in this process.

Example

Reduction in mechanical energy due to

work done by friction:

Rearranging equation:

where

Wf is the work done by the book, thus it

should be a negative value.


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Work-Energy Theorem

The net work done W on an object is equal to the change in the object’s kinetic energy ΔK.

To be more precise, the only change in the system is in its speed (not velocity).

22

2

1

2

1if mvmvW

Example

vf > vi vf < vi

Additional Knowledge: Conservation of Energy vs Work-Energy Theorem

Total (net) work done Wnet is the sum of the work done by conservative force WC, and the work

done by non-conservative force WNC.

NCCnet WWW

According to work-energy theorem,

KEWnet

Then,

KEWW NCC

The concept of potential energy is associated only with a type of force known as a “conservative”

force. The change in the potential energy of the system equals the negative of the work done by that force (work done by the system). Thus,

PEWC

Substitute equation (2) into (1):

PEKEWNC

If no non-conservative force acting on the object:

ifif PEPEKEKEPEKE 0

ffii PEKEPEKE

If conservative such as frictional force acting on the object:

ififf PEPEKEKEPEKEW

fffii PEKEWPEKE

(1)

(2)

Conservation of Energy

Conservation of Energy

with Non-Conservative

force


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5.3 Power

Average power Instantaneous power

Average power �̅� is the average rate at

which work is done, and it is obtained by

dividing ΔW by the time required to

perform the work Δt.

t

WP

Power is a scalar quantity.

SI unit of power is J s-1

or Watt (W).

The instantaneous power is then the limiting value of

the average power as the time interval Δt approaches zero.

vFvF

dt

dsF

dt

dW

t

WP

t

cos

coslim

0

Explanation:

It is the power of any object at an instant.

Additional Knowledge: Horsepower

One metric horsepower is needed to lift 75 kilograms by 1 meter in 1 second.

Example: Input power and output power

A motor is supplied with 2500 J of energy in 15 minutes to lift a 25 kg load to a height of 6 m.

Power input (power supplied to

the machine)

E = 2500 J

t = 15 minutes = 900 s

W78.2t

EP

Power output (useful power

produced)

m = 25 kg

h = 6 m

W64.1t

mgh

t

EP

Difference between power input and

power output is due to power lost

(dissipate) due to friction, heat, etc…


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Exercise

Work

1. The brakes of a truck cause it to slow down by applying a retarding force (force that oppose

the motion) of 3.0×103 N to the truck over a distance of 850 m. What is the work done by

this force on the truck? Is the work positive or negative? Why?

2. 1200-N crate rests on the floor. How much work is required to move it at constant speed

i. 5.0 m along the floor against a friction force of 230 N?

ii. 5.0 m vertically?

3. A farmer hitches his tractor to a sled loaded with

firewood and pulls it a distance of 20 m along

level ground. The total weight of sled and load is

14.7 N. The tractor exerts a constant 5000 N force

at an angle of 36.9° above the horizontal. There is

a 3500 N friction force opposing the sled’s

motion. Find the work done by each force acting

on the sled and the total work done by all forces.

4. A force acts on a particle varies with x as shown in figure

below. Calculate the work done by the force on the particle as it moves from x = 0 to x = 6.0 m.

5. A horizontal force F is applied to a 2.0 kg

radio-controlled car as it moves along a

straight track. The force varies with the

displacement of the car as shown in the

figure. Calculate the work done by the force F when the car move from s = 0 to s = 7 m.

Energy and Conservation of Energy

1. How much work must be done to stop a 925 kg car traveling at 95 km h-1

?

2. A woman weighing 600 N steps on a bathroom

scale containing a stiff spring. In equilibrium,

the spring is compressed 1.0 cm under her

weight. Find the force constant of the spring

and the total work done on it during the

compression.

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3. A gymnast springs vertically upward from a

trampoline as in Figure below. The gymnast leaves

the trampoline at a height of 1.20 m and reaches a

maximum height of 4.80 m before falling back down.

All heights are measured with respect to the ground.

Ignoring air resistance, determine the initial speed v0

with which the gymnast leaves the trampoline.

4. The space probe Deep Space 1 was launched October 24, 1998, and it used a type of engine

called an ion propulsion drive. An ion propulsion drive generates only a weak force (or

thrust), but can do so for long periods of time using only small amounts of fuel. Suppose the

probe, which has a mass of 474 kg, is traveling at an initial speed of 275 m s-1

. No forces act

on it except the 5.60×10-2

N thrust of its engine. The displacement travelled by the probe is

2.42×109 m. Determine the final speed of the probe, assuming that its mass remains nearly

constant.

5. Assuming the height of the hill in Figure below is

40 m, and the roller-coaster car starts from rest at

the top, calculate

i. the speed of the roller-coaster car at the bottom

of the hill

ii. at what height it will have half this speed Take y = 0 at the bottom of the hill.

6. A 66 kg skier starts from rest at the top of a 1200 m long trail which drops a total of 230 m

from top to bottom. At the bottom, the skier is moving 11.0 m s-1

. How much energy was dissipated by friction?

Power

1. A 50.0 kg marathon runner runs up the stairs to the top of Chicago’s 443.0 m tall Sears

Tower, the tallest building in the United States. To reach the top in 15.0 minutes, what must

be her average power output in watts?

2. Calculate the power required of a 1400 kg car under the following circumstances:

i. the car climbs a 10° hill at a steady 80 km h-1

ii. the car accelerates along a level (flat) road from 90 km h-1

to 110 km h-1

in 6.0 s

Assume the average frictional force on the car is 700 N throughout.


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3. What minimum power must a motor have to be able to drag a 370 kg box along a level floor

at a speed of 1.20 m s-1

if the coefficient of friction is 0.45?

4. The motor of a ski boat generates an average power of 7.50 ×104 W when the boat is moving

at a constant speed of 12 m s-1

. When the boat is pulling a skier at the same speed, the engine

must generate an average power of 8.30 ×10 4 W. What is the tension in the tow rope that is

pulling the skier?

HOT Questions

1. A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before

striking a vertical coiled spring, which it compresses an amount of Y = 15.0 cm as shown in

Figure below. Determine the spring stiffness constant k of the spring. Assume the spring has

negligible mass, and ignore air resistance. Measure all distances from the point where the

ball first touches the uncompressed spring (y = 0 at this point)

2. The roller-coaster car in Figure below reaches a

vertical height of only on the second hill, where it

slows to a momentary stop. It travelled a total

distance of 400 m. Determine the thermal energy

produced and estimate the average friction force

(assume it is roughly constant) on the car, whose

mass is 1000 kg.

Hint: Friction force is a non-conservative force; it

depends on the path travelled by the object. Wf = fd, where d is the distance travelled.

3. A pump is required to lift 800 kg of water per minute from a well 14.0 m deep and eject it

with a speed of 18.0 m s-1

.

i. How much work is done per minute in lifting the water?

ii. How much work is done in giving the water the kinetic energy it has when ejected?

iii. What must be the power output of the pump?

chapter 5 work, energy and power

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