Work and Energy

Work Done by a Constant Force

In physics, work is described by what is accomplished when a force acts on an object, and the object moves through a distance.

The work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement.

W = F! d = Fd cos!F is the magnitude of the constant force, d is the magnitude of the displacement of the object, and ! is the angle between the directions of the force and the displacement. The cos ! factor appears because F cos ! (= F||) is the component of

F that is parallel to d. Work is a scalar quantity—it has no direction, but only magnitude, which can be positive or negative.

Work Done by a Constant Force In the case in which the motion and the force are in the same direction, we have ! = 0 and cos ! = 1, so W = Fd.

The unit of work in SI system is measured in Newton-meters (N.m). The special name given to this unit is Joule (named after British physicist, James Prescott Joule) defined as the distance moved multiplied by the component of the force in the direction of displacement.

1 J = 1 N.m

NOTE: A force can be exerted on an object and yet do no work. If you hold a heavy bag of groceries in your hands at rest, you do no work on it. You do exert a force on the bag, but the displacement of the bag is zero, so the work done by you on the bag is W = 0. You need both a force and a displacement to do work.

Example (1) A person pulls a 50-kg crate 40 m along a horizontal floor by a constant force FP = 100 N, which acts at a 37° angle as shown in the figure. The floor is rough and exerts a friction force Ffr = 50N. Determine (a)! the work done by each force acting on the crate, and (b)! the net work done on the crate.

Work Done by a Constant Force

Work done by forces that oppose the direction of motion, such as friction, will be negative.

Centripetal forces do no work, as they are always perpendicular to the direction of motion.

W =!F !!d = Fd cos"

Example (2) Does the Earth do work on the Moon? The Moon revolves around the Earth in a nearly circular orbit, kept there by the gravitational force exerted by the Earth. Does gravity do (a)! positive work, (b)! negative work, (c)! no work on the Moon

Work Done by a Varying Force If the force acting on an object is constant, the work done by that force will be

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. As the pieces become very narrow, the work done is the area under the force vs. distance curve.

W =!F !!d = Fd cos"

The work done by a variable force in moving an object between two points is equal to the area under the F|| vs. d curve between those two points.

Kinetic Energy, and the Work-Energy Principle

In a traditional way, energy is defined as “the capacity of doing work”. This definition is valid for mechanical energy, however is not always applicable (for example for the energy associated with heat).

To obtain a quantitative definition for kinetic energy, consider a simple rigid object of mass m that is moving in a straight line with an initial speed v1. To accelerate it uniformly to a speed v2, a constant net force Fnet must be exerted on it parallel to its motion over a displacement d.

An object in motion has the ability to do work and thus can be said to have energy.

Kinetic Energy, and the Work-Energy Principle

The net work done on the object is Wnet = Fnet d. We apply Newton's second law, Fnet = ma, and use v2

2 – v12 = 2ad

F = ma

v22 ! v1

2 = 2ad" a = v22 ! v1

2

2d

#$%

&%

'(%

)%" F = m v2

2 ! v12

2d

Wnet = Fnetd = ma( )d = m v22 ! v1

2

2d"#$

%&'d = m v2

2 ! v12

2"#$

%&'= 12m v2

2 ! v12( ) = 12mv2

2 ! 12mv1

2

Wnet = KE2 ! KE1 = "KE = 12mv2

2 ! 12mv1

2

The quantity ! mv2 is defined as the translational kinetic energy (KE) of the object.

Kinetic Energy, and the Work-Energy Principle

Work-energy principle: The net work done on an object is equal to the change in the object's kinetic energy.

Wnet = !KE = 12mv2

2 " 12mv1

2

If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases.

Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules.

Potential Energy

An object can have kinetic energy by virtue of its motion and can have potential energy by virtue of its surroundings.

Familiar examples of potential energy:

In general the potential energy of an object is the capacity or potential of the object to do work even though it is not yet actually doing it. From these definitions and examples, it can be concluded that: the energy can be stored, for later use, in the form of potential energy.

-- A wound-up spring -- A stretched elastic band -- An object at some height above the ground

In general, the change in potential energy associated with a particular force is equal to the negative of the work done by that force when the object is moved from one point to a second point .

Gravitational Potential Energy

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

Wext = Fextd cos0! = mgh = mgy2 !mgy1 = mg y2 ! y1( )

WG = FGd cos! = mghcos180! = "mgh = "mg y2 " y1( )

Gravity is also acting on the object as it moves from y1 to y2, and does the work on the object equal to:

We therefore define the gravitational potential energy of an object, due to Earth's gravity, as the product of the object's weight mg and its height y above some reference level (such as the ground).

PEG = mgy

Wext = mg y2 ! y1( ) " Wext = PE2 ! PE1 = #PEGWG = !mg y2 ! y1( ) " WG = ! PE2 ! PE1( ) = !#PEG

!"!#PEG =Wext

#PEG = $WG

%&'

Example (3) 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.

Example (4)

By what factor does the kinetic energy of a car change when its speed is tripled?

a) no change at all b) factor of 3 c) factor of 6 d) factor of 9 e) factor of 12

Potential Energy of Elastic Spring Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

The force required to compress or stretch a spring is:

where k is called the spring constant, and needs to be measured for each spring.

Fext = kx

FS = !kx

The spring itself exerts a force Fs in the opposite direction on the hand,

This equation is known as spring equation and also as Hook’s Law.

Potential Energy of Elastic Spring

Fext = kx

Wext = Fx =12kx!

"#$%& x( ) = 1

2kx2

PEel =12kx2

The force increases as the spring is stretched or compressed further.

Conservative and Non-conservative Forces

Forces such as gravity, for which the work done does not depend on the path taken but only on the initial and final positions, are called conservative forces. Examples of Conservative Forces: Gravitational, Elastic, Electric

Many forces, such as friction and a push or pull exerted by a person, are non-conservative forces since any work they do depends on the path. Examples of Non-conservative Forces: Friction, Air Resistance, Tension in Cord, Motor or Rocket Propulsion, Push or Pull by a Person

Conservative and Non-conservative Forces

Potential energy can be defined only for a conservative force. Thus, although potential energy is always associated with a force, not all forces have a potential energy. For example, there is no potential energy for friction.

Therefore, we distinguish between the work done by conservative forces and the work done by non-conservative forces.

We can extend the work-energy principle to include potential energy and write the total (net) work Wnet as a sum of the work done by conservative forces, WC, and the work done by non-conservative forces, WNC:

Wnet =WC +WNC = !KE WNC = !KE "WC

WC = "!PE

#$%

&%' WNC = !KE + !PE

The work WNC done by the non-conservative forces acting on an object is equal to the total change in kinetic and potential energies.

Mechanical Energy and Its Conservation

!KE + !PE = 0KE2 " KE1( ) + PE2 " PE1( ) = 0

#$%

&%' KE2 + PE2 = KE1 + PE1

If there are no non-conservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero—the kinetic and potential energy changes are equal but opposite in sign. This allows us to define the total mechanical energy:

E = KE + PE

When no non-conservative forces do work, then WNC = 0 in the general form of the work-energy principle. Then for conservative forces we have:

KE2 + PE2 = KE1 + PE1

E = KE + PE!"#

$ E2 = E1 = constantConservative forces only

Problem Solving Using Conservation of Mechanical Energy

A simple example of the conservation of mechanical energy (neglecting air resistance) is a rock allowed to fall due to Earth's gravity from a height h above the ground. At any point along the path, the total mechanical energy is given by

E = KE + PE = 12mv2 +mgy

The total mechanical energy at point1is equal to the total mechanical energy at point2

12mv1

2 +mgy1 =12mv2

2 +mgy2

Other Forms of Energy & Energy Transformation; The Law of Conservation of Energy

Besides the kinetic energy and potential energy of mechanical systems, other forms of energy can be defined as well. These include electric energy, nuclear energy, thermal energy, and the chemical energy stored in food and fuels. These other forms of energy are considered to be kinetic or potential energy at the atomic or molecular level.

Accounting for all forms of energy, we find that the total energy neither increases nor decreases. Energy as a whole is conserved. The law of conservation of energy, one of the most important principles in physics; can be stated as:

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.

Energy Conservation with Dissipative Forces

In many situations non-conservative forces, such as frictional forces reduce the mechanical energy (but not the total energy), they are called dissipative forces.

WNC = !KE + !PEWNC = KE2 " KE1 + PE2 " PE1

#$%& KE1 + PE1 +WNC = KE2 + PE2

KE1 + PE1 +WNC = KE2 + PE2WNC = !Ffrd

"#$%

12 mv1

2 +mgy1 ! Ffrd = 12 mv2

2 +mgy2KE1 + PE1 = KE2 + PE2 + Ffrd

&'(

)(

Power Power is defined as the rate at which work is done.

P = Average Power = worktime

= enrgy transformedtime

In SI units, power is measured in Joules per second, and this unit is given a special name (named after James Watt, Scottish Engineer), the watt (W). 1 W = 1 J/s.

In the British system, the unit of power is the foot-pound per second (ft ! lb/s). For practical purposes, a larger unit is often used, the horsepower. One horsepower (hp) is defined as 550 ft ! lb/s, which equals 746 W. An engine's power is usually specified in hp or in kW.

P = Wt

Js( ) Also! "!! P = W

t= Fd

t= F v

Example (5)

A roller-coaster car starts from rest at the top of the hill. The height of the hill is 40m. Calculate (a)! the speed of the roller-coaster car at the bottom of the hill (b)! at what height it will have half this speed. Take y=0 at the bottom of the hill

Example (6)

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 y=15.0 cm. 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).

Example (7)

Calculate the power required of a 1400-kg car under the following circumstances: (a) the car climbs a 10° hill (a fairly steep hill) at a steady 80 km/h, (b) the car accelerates along a level road from 90 to 110 km/h in 6.0 s to pass another car. Assume the average retarding force on the car is FR = 700 N throughout.