Chapter 7 - Work and Energy

• Work– Definition of Work [units]

– Work done by a constant force (e.g friction,weight)

– Work done by a varying force (e.g. a spring)

– Work in 3 dimensions – General Definition

• Work and Kinetic Energy– Definition of Kinetic Energy

– Work-Energy Principle

Definitions

• Work - The means of transferring energy by the application of a force.

• Work is the product of the magnitude of displacement times the component of that force in the direction of the displacement.

• Work is a scalar • Energy - The state of one

or more objects. A scalar quantity, it defines the ability to do work.

//W F r F r cos

UnitsPhysicalQuantity

DimensionSymbol

SI MKS SI CGS USCustomary

Length [L] m cm ft

Mass [M] kg g slug

Time [T] sec sec sec

Acceleration [L/T2] m/s2 cm/s2 ft/s2

Force [M-L/T2] newton (N)kg-m/s2

Dyneg-cm/s2

pound (lb)slug- ft/s2

Energy [M-L2/T2] Joule (J)N-m

kg-m2/s2

ErgDyne-cmg-cm2/s2

Ft-lbslug-ft2/s2

Problem 1

• A 1500 kg car accelerates uniformly from rest to a speed of 10 m/s in 3 s.

• Find the work done on the car in this time

//W F r F r cos

How much work is done by this guy?

Walking at a constant speed

//W F r F r cos

r

Problem 3

• m = 50 kg

• displacement = 40 m

• force applied = 100 N

• 37o angle wrt floor

• k = 0.1

• Find net work done moving the crate

Vector Multiplication – Scalar Product

A B A B cos

ˆ ˆ ˆ ˆ ˆ ˆi i j j k k 1

ˆ ˆ ˆ ˆ ˆ ˆi j i k j k 0

x y zˆ ˆ ˆA A i A j A k

x y zˆ ˆ ˆB B i B j B k

x x y y z zA B A B A B A B

A more elegant definition for work

//W F r F r cos

A B A B cos

W F r

Problem 4

• How much work is done pulling the wagon 100 m in the direction shown by the boy applying the force:

ˆ ˆF 17Ni 10Nj

r

Work done by a varying force

1 1 1 1W F cos l 7

i i ii 1

W F cos l

i

7 b b

i i i a al 0i 1

W lim F cos l Fcos dl F dl

Work in three dimensions

x y zˆ ˆ ˆF F i F j F k

ˆ ˆ ˆdr dxi dyj dzk

b b b

a a a

b x y z

x y za x y zW F dr F dx F dy F dz

Problem 5

5 10 15 x (m)

3

2

1

Fx (N)

How much work is done by this force?

Hooke’s Law and the work to compress/extend a spring

sF kx

b

a

b x

xa xW F dr F dx

b

a

x x 2P x 0

1W kx dx kx

2

Kinetic Energy and the Work-Energy Principle

2 22 20

0

v v 1 1W F d m a d m d mv mv

2d 2 2

0W K K K

21K mv

2

And you can show this with calculus too!

b

a

b x

xa xW F dr F dx

2 2 2 2 22 11 1 1

dv dx 1 1W m dx m dv mvdv mv mv

dt dt 2 2

Problem 6

• A 3 kg mass has an initial velocity, v = (5i - 3j) m/s.• What is the kinetic energy at this time?• The velocity changes to (8i + 4j) m/s.• What is the change in kinetic energy?• How much work was done?

Problem 7

• A 2 kg block is attached to a light spring of force constant 500 N/m. The block is pulled 5 cm to the right and of equilibrium. How much work is required to move the block?

• If released from rest, find the speed of the block as it passes back through the equilibrium position if– the horizontal surface is frictionless.– the coefficient of friction is 0.35.