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Work and Energy

- Work, Kinetic Energy, Potential Energy, and

Conservation of Energy

Work Done by a Constant Force

Work is the product of the displacement and the

force component in the direction of the

displacement W F cosq d (a

scalar) Where the units of work are the Joule,

J, 1 J 1 N m

Example of a constant force work calculation.

F 250 N at 300 below horizontal. D 6.65 m

parallel to ground. How much of F is parallel to

ground F cos (300) 250 cos(300) 216.5

N, So the work done by this force is, W F

cosq d 216.5 N x 6.65 m 1440 Nm 1440 J

Work Done by a Variable Force

- A variable force is one that depends upon a

variable. For example, the drag force that acts

against moving objects depends upon the velocity

of the object. - An example of a variable force is the force it

takes to extend a spring - F - k x,
- Where k is the spring constant.

Hookes Law

A linear restoring force. The spring stores

energy and can return it by doing work. The

amount of work, W, is W ½ k x2

F lkxl

F, N

X, m

Example

lFl 60 N/m x, where x goes from 0 to 0.15 m.

How much work does this force do? What is k?

k 60 N/m What is the definition of

work done by a spring? W ½ kx2 ½ (60

N/m)(0.15m)2 0.675 J

Work-Energy Theorem

Since F ma, and W Fcosq d, then work must

result in an objects velocity being

altered. Recall, v2 v02 2 a x , multiply

by m, m v2 mv02 2 ma x, divide by 2 ½ mv2

½ mv02 F x, identify Fx as W ½ mv2 ½

mv02 W, define K ½ mv2 So, K K0 W, or

DK W

Example

How fast will a 200 g object be moving after

being pushed by a 75 N over a distance of 60

cm? F 75 N, v0 0 m/s, v ?, d 0.60 m, m

0.20 kg. K ? What is initial kinetic energy, K0

So, DK K W Fd 75 N x 0.60 m 45 J. If

K 45 J ½ mv2, the v2 2(45J)/(0.20kg)450

m2/s2, So v 21 m/s.

Another Example

If it takes 5000 J to take a car from 10 kph to

20 kph, how much work will it take to go from 20

kph to 30 kph? Given W 5000 J, when v0 10

km/h and v 20 km/h. W ? When v0 20 km/h to

v 30 km/h? Equation W DK ½ mv2 ½ mv02

Solution 5000 J ½ m (20,000 m/3600 s)2 ½ m

(10,000 m/3600 s)2 Solve for mass and use in

second calculation W ½ m (30,000 m/3600 s)2

½ m (20,000 m/3600 s)2 W J

Potential Energy

- Energy that is associated with position in a

force field. - Examples
- Gravitational Potential Energy
- Spring Potential Energy
- U, the symbol for potential energy.
- Units for PE are Joules, J

Potential Energy-Zero Point

The amount of PE is calculated relative to a zero

point. For the spring the zero point is the

equilibrium length of the spring. U ½ kx2 ½

kx02 For gravity, the zero point is ground

level. U mgy mgy0

Example

Yattic 4.5, y0 0, ybase -3.0, m 1.5 kg

4.5m

Ua 1.5 kg 9.8 m/s2 4.5 m

U01.5kg9.8m/s2 0m 0 J

0

Ub1.5kg9.8m/s2-3 m

-3.0 m

Example

How much would you have to stretch a k 45 N/m

spring to store 1 J of energy? U 1 J, k 45

N/m, x ???? U ½ k x2 X 2 U/k1/2 2 ( 1

J)/ ( 45 N/m)1/2 0.21 m 21 cm

Conservation of Energy

Total Mechanical Energy, E K U. In physical

phenomena, total mechanical energy is conserved,

Ef Eo Kf Uf Ko Uo

½ mvf2 Uf ½ mvo2 Uo

Energy and Non-conservative Forces

Since experience and experiment have validated

the conservation of energy principle in all

cases, we know that when it appears to fail, it

only means that we have not accounted for all the

energy and work contributions.

Continued

- What if we find,
- Ef E0 gt 0 ?
- Then, some of the energy that was available

initially has not manifest itself in the final

state. - E0 Ko Uo and Ef Kf Uf , so
- Kf Uf (Ko Uo) gt0
- Kf - Ko Uf - Uo gt0

Continued

- DK DU gt0
- This means that a non-conservative force has done

work in the problem and consumed some of the

energy. - Wnc Example friction when working against

friction, the path taken makes a difference.

Power- the rate of doing work

P, symbol for power. P W/t, so units are J/s

Watt (W) Alternate expressions Since W Fd,

then P W/t Fd/t Fv Efficiency What you get

out versus what you put in e Wout/Win

Pout/Pin

Problem

A 3250 kg plane reaches a cruising altitude of

10,000 m moving at 850 km/h in 12.5 minutes. If

the jet engines supply 1500 hp, what is the

efficiency of the plane? M 3250 kg, t 12.5

min 750 s, y 10,000 m V 850 km/h 236 m/s,

Pin 1500 hp 1119000 Watts.

Continued

e P(out)/P(in) W(out)/t / 1119000 Watts e

(DKDU)/t/1119000 (½ mv2

mgy)/t/1119000 Watts (0.53250kg(236m/s)2

3250kg9.8m/s210000m)/750/1119000 Watts

(90506000318500000)750/1119000 0.48 or 48

(No Transcript)

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