Work and Energy• Work• Kinetic Energy• Work – Energy Theorem• Gravitational Potential Energy• Mechanical Energy• Conservation of Energy

StarterThe angle between two vectors is found by placing their tailstogether and picking the smallest angle between them. For example, the angle between B and D is 45 degrees. Find the angle between:

1. A and B

2. B and C

3. C and D

4. A and E1. 180o 2. 90o 3. 135o 4. 90o

Practice - Work

Work = Frcos(q) Units = Joules

F = magnitude of the force

r = magnitude of the displacement

q = the angle between F and r

Example

F = 100N r = 2m q = 30 degrees

A 100N force acts on a box at 30 degrees with respect to the horizon,moving the box 2m to the right. How much work did the force do?

Work = Frcos(q) = 100(2)cos30 = 173J

Example

Find the total work done on this box as it moves 3m to the right.Work done by P: W = 200(3)cos(0) = 600J

Work done by N: W = Nrcos(90) = 0Work done by f : W = 80(3) cos(180) = -240JWork done by Weight : W = 100(3)cos(90) = 0

Total Work = 600 + 0 + (-240) + 0 = 360J

Kinetic Energy – Energy of Motion

KE = (1/2)mv2

Example: Find the kinetic energy of a 20kg mass moving at 10 m/s.

KE = (1/2)mv2 = (1/2)(20)(10)2 = 1000J Example: If you double your velocity, what happens to the KE?

KE (new) = (1/2)m(2v)2 = 4KE(original)

Work Energy Theorem

Work Total = ½mvf2 – ½mvi

2

or

Work Total = DKE

Example

If this box starts at rest, what is its velocity after it moves 3m?

Previously we saw that the total work was 360J. Work Total = (1/2)mvf

2 – (1/2)mvi2

360 = (1/2)(10)vf2

vf = 8.48 m/s

ExampleIf a mass is dropped from rest 20m above the ground, how fast is it moving when it hits the ground?

Work Total = (1/2)mvf2 – (1/2)mvi

2 = (1/2)mvf2

Work = Frcosq = mgrcos(0) = mgr = (1/2)mvf2

vf = [ 2gr ] ½ = [ 2(9.8)(20) ] ½ = 19.8 m/s

Starter #2At the top of her swing, 2 meters from the ground, a child has a potential energy of 500 Joules. When she is 1m from the ground, her kinetic energy is

a. 500Jb. 300Jc. 250Jd. 150J.

StarterA golf ball moving at 40 m/s has a kinetic energy of 35 Joules. What is its kinetic energy if it instead moves at 120 m/s?

a. 75Jb. 105Jc. 315Jd. None of these.

Energy TypesEnergy of Motion – Kinetic Energy

Stored Energy – Potential Energy

Gravitational Potential Energy

The work you do to lift an object is equal to its increase in gravitational PE. PE = mgh

h = vertical distance from ground level.

ExampleHow much work does a 100kg person do walking up a flight of stairs that takes her 10m off of the ground?

Work = increase in PE = mgh = 100(9.8)10 = 9800 Joules.

(Note: 1 Calorie = 4128J, so this person burned a little over 2 calories. )

( Note: 1 taco = 200 Calories = 825,600 Joules)

Total Mechanical Energy, E

E = KE + PEIf there is no friction, the total mechanical energy is conserved.This means KE + PE is always the same number.

So if KE drops, PE must rise. If PE drops, KE must rise.

Another way to say this, is that KE is converted to PE, or PE is converted to KE.

Conservation of Energy

Example A 1 kg rock is dropped from rest from

a building 20m high. Fill in the table.

Height KE PE E20      10      5      0      

PE at the top is mgh =1(9.8)(20) = 196J

196JKE at the top is zero.

0

E at the top (and everywhere)is 0 + 196J = 196J

196J

196J

196J

196J

PE at the bottom is zero…………

0196J

98J

49J

98J

147J

SummaryWork = Frcos(q) KE =

(1/2)mv2

Work Total = ½mvf2 –

½mvi2

or Work Total = DKE PE = mgh E = KE + PE

StarterA 10kg rock is moving at 10m/s and is 10m above the ground. What is its total energy, E?

KE = (1/2)mv2 = (.5)(10)(102) = 500J

PE = mgh = 10(9.8)10 = 980J

E = PE + KE = 1480J