Work, Energy, and PowerWork, Energy, and Power

Samar Hathout

KDTH 101

Work is the transfer of energy through motion. In order for work to take place, a force must be exerted through a distance. The amount of work done depends on two things: the amount of force exerted and the distance over which the force is applied. There are two factors to keep in mind when deciding when work is being done: something has to move and the motion must be in the direction of the applied force. Work can be calculated by using the following formula: Work=force x distance

WorkWork

Work is done on the books when they are being lifted, but no work is done on them when they are being held or carried horizontally.

WorkWork

Work can be positive Work can be positive or negativeor negative

• Man does positive work lifting box

• Man does negative work lowering box

• Gravity does positive work when box lowers

• Gravity does negative work when box is raised

Work done by a constant ForceWork done by a constant Force

Ekin = Wnet

• W = F s = |F| |s| cos = Fs s

|F| : magnitude of force |s| = s : magnitude of displacement Fs = magnitude of force in direction of displacement :

Fs = |F| cos

: angle between displacement and force vectors• Kinetic energy : Ekin= 1/2 m v2

• Work-Kinetic Energy Theorem:

F

s

Work Done by GravityWork Done by Gravity

Example 1: Drop ball

Yi = h

Yf = hf

Wg = (mg)(S)

S = h0-hf

Wg = mg(h0-hf)

= mg(h0-hf)

= Epot,initial – Epot,final

mgS

y

x

Yi = h0

mgS

y

x

Work Done by GravityWork Done by Gravity

Example 2: Toss ball up

Wg = (mg)(S)

S = h0-hf

Wg =-mg(h0-hf)

= Epot,initial – Epot,final

Yi = h0

Yf = hf

mgS

y

x

Work Done by GravityWork Done by Gravity

Example 3: Slide block down incline

Wg = (mg)(S)cos

S = h/cos

Wg = mg(h/cos)cos

Wg = mgh

with h= h0-hf

h

mgS

Work done by gravity is independent of path

taken between h0 and hf

=> The gravitational force is a conservative force.

h0

hf

Work done by a Variable ForceWork done by a Variable Force

The magnitude of the force now depends on the

displacement: Fs(s)

Then the work done by this force is equal to the

area under the graph of Fs versus s, which can be

approximated as follows:

W = Wi Fs(si) s = (Fs(s1)+Fs(s2)+…) s

Concept QuestionConcept Question

Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball reach the bottom with the highest speed?

1. Dropping2. Slide on ramp (no friction)3. Swinging down4. All the same

In all three experiments, the balls fall from the same height and therefore the same amount of their gravitational potential energy is converted to kinetic energy. If their kinetic energies are all the same, and their masses are the same, the balls must all have the same speed at the end.

1 2 3correct

Kinetic Energy Potential Energy

Types of Energy

Radiant ElectricalChemical

Thermal Nuclear

Magnetic

Sound

Mechanical

Forms of Energy

Mechanical energy is the movement of machine parts. Mechanical energy is also the total amount of kinetic and potential energy in a system. Wind-up toys, grandfather clocks, and pogo sticks are examples of mechanical energy. Wind power uses mechanical energy to help create electricity.

Potential energy + Kinetic energy = Mechanical energy

Mechanical EnergyMechanical Energy

Potential energy + Kinetic energy = Mechanical energy

Example of energy changes in a swing or pendulum.

Mechanical EnergyMechanical Energy

Conservation of Mechanical EnergyConservation of Mechanical Energy

Total mechanical energy of an object remains constant provided the net work done by non-conservative

forces is zero: Etot = Ekin + Epot = constantor

Ekin,f+Epot,f = Ekin,0+Epot,0

Otherwise, in the presence of net work done bynon-conservative forces (e.g. friction):

Wnc = Ekin,f – Ekin,0 + Epot,f-Epot,i

Example ProblemExample Problem

Suppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces? 1. 0J 2. 50J 3. -50J 4. 225J 5. -225J

correct

Work done by non-conservative forces equals the difference between final and initial kinetic energies plus the difference between the final and initial gravitational potential energies.

W = (300-75) + ((-25) - 250) = 225 - 275 = -50J.

Samar HathoutSamar Hathout

Kinetic EnergyKinetic Energy

Same units as work

Remember the Eq. of motion

Multiply both sides by m,

1

2mv f

2 1

2mvi

2 max

KE f KEi Fx

v f2

2vi2

2ax

KE 1

2mv2

Samar Hathout

ExampleExample

Samar Hathout

Potential EnergyPotential Energy

Potential energy exists whenever an object which has mass has a position within a force field (gravitational, magnetic, electrical).We will focus primarily on

gravitational potential energy (energy an object has because of its height above the Earth)

Potential EnergyPotential Energy

If force depends on distance,

For gravity (near Earth’s surface)

PE Fx

PE mgh

Samar Hathout

Conservation of EnergyConservation of Energy

Conservative forces:• Gravity, electrical, QCD…Non-conservative forces:• Friction, air resistance…Non-conservative forces still conserve energy!Energy just transfers to thermal energy

PE f KE f PEi KEiKE PE

Samar Hathout

ExampleExample

A diver of mass m drops from a board 10.0 m above the water surface, as in the Figure. Find his speed 5.00 m above the water surface. Neglect air resistance.

9.9 m/s

ExampleExample

A skier slides down the frictionless slope as shown. What is the skier’s speed at the bottom?

H=40 m

L=250 m

start

finish

28.0 m/s

Example Example

Three identical balls are thrown from the top of a building with the same initial speed. Initially, Ball 1 moves horizontally. Ball 2 moves upward. Ball 3 moves downward.

Neglecting air resistance, which ball has the fastest speed when it hits the ground?A) Ball 1

B) Ball 2C) Ball 3D) All have the same speed.

Springs (Hooke’s Law)Springs (Hooke’s Law)

Proportional to displacement from equilibrium

F kx

Potential Energy of Potential Energy of SpringSpring

PE=-Fx

x

F

PE 1

2(kx)x

PE 1

2kx2

x

Example Example

b) To what height h does the block rise when moving up the incline?

A 0.50-kg block rests on a horizontal, frictionless surface as in the figure; it is pressed against a light spring having a spring constant of k = 800 N/m, with an initial compression of 2.0 cm.

3.2 cm

Power Power

Average power is the average rate at which a net force

does work:

Pav = Wnet / tSI unit: [P] = J/s = watt (W)

Or Pav = Fnet s /t = Fnet vav

Example Example

A 1967 Corvette has a weight of 3020 lbs. The 427 cu-in engine was rated at 435 hp at 5400 rpm.a) If the engine used all 435 hp at 100% efficiency during acceleration, what speed would the car attain after 6 seconds?b) What is the average acceleration? (in “g”s)

a) 120 mph b) 0.91g

Example Example

Consider the Corvette (w=3020 lbs) having constantacceleration of a=0.91ga) What is the power when v=10 mph?b) What is the power output when v=100 mph?

a) 73.1 hp b) 732 hp (in real world a is larger at low v)