• The concept of energy (and the conservation of energy – chapter 8) is one of the most important topics in physics.

• Work• Kinetic energy• Energy approach (“system model”) is often simpler than Newton’s

second law (“particle model”).• Scalar product (dot product); work is scalar product of force and

displacement. • Hooke’s law, springs• Power

Chapter 7: Work; Energy of a System

Reading assignment: Chapter 7

Homework 7.1 (due Th. Oct. 4): CQ1, QQ1, QQ2, AE2, 1, 2, 9, 11, 12

Homework 7.2 (due Tu. Oct 9.): CQ5, 14, 15, 17, 25, 27, 31, 33, 42, 43

Remember: Homework 6 is due Wednesday, Oct. 3

Work (as defined by a physicist)

Definition:

The work done on an object by an external force is:

the product of the component of the force in the direction of the displacement and the magnitude of the displacement.

cos dFW

How much work is done when just holding up an object?

W

dFW cos

How much work is done when the displacement is perpendicular to the force?

W

dFW cos

What is the work done when lifting?(at constant speed)

- By the applied force?

- By the gravitational force?

W

dFW cos

Sign convention:

W positive:

If F and d are parallel.

If energy is transferred into the system.

W negative:

If F and d are antiparallel.

If energy is transferred out of the system.

How much work is the strongest man doing when he lifts a 100 kg boulder by 1 m.

A.0 J

B.9800 J

C.9.8 J

D.100 J

E.None of the above

Work is a scalar quantity (not a vector).

Work has units of

Newton·meter (N·m) = the Joule (J)

Born: Dec. 24 1818, Salford, Lancashire, England

Died: Oct. 11, 1889

Joule studied the nature of heat and discovered its relationship to mechanical work. This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics.

(from Wikipedia)

Black board example 7.1

A donkey is pulling a cart with a force of magnitude F = 500 N at an angle of 30º with the horizontal. The cart is moving at constant velocity.

1.Complete the free body diagram.

500 N

2. Calculate the work done (a) by the donkey (applied force) (b) gravity, (c) normal force, (d) frictional force, as the cart is pulled for one mile (1609 m).

cos dFdFW

Work is the scalar product (or dot product) of the force F and the displacement d.

F and d are vectors

W is a scalar quantity

Definition of dot product and work

Scalar product between vector A and B

cos ABBA

ABBA

Scalar product is commutative:

CABACBA

)(

Definition 1:

Distributive law of multiplication:

Scalar Product using unit vectors:

kBjBiBBkAjAiAA zyxzyx

We have the vectors A and B:

Then, defintion 2:

zzyyxx BABABABA

2AAAAAAAAA zzyyxx

Black board example 7.2

A particle moving in the x-y plane undergoes a displacement d = (2.0i + 3.0j) m as a constant force F = (5.0i + 2.0j) N acts on the particle. Calculate

(a) The magnitude of the displacement and the force.

(b) The work done by F.

(c) The angle between F and d.

What if the force varies? We have to integrate the force along x

f

i

x

x

dxxFWWork done by a varying force:

Thus, the work is equal to the area under the F(x) vs. x curve.

Black board example 7.3

i-clicker

A force acting on a particle varies as shown in the Figure.

What is the work done on the particle as it is moved from x = 0 to x = 6 m? (Hint: It is the area under the curve.)

A. 5 J

B. 10 J

C. 20 J

D. 25 J

E. 30 J

Consider a spring

Hooke’s law:(Force required to stretch or compress a spring by x):

xkFs

k is the spring constant of a spring.

Stiff springs have a large k value.

)(2

1 22fi xxkW

Work done by a spring

xi xf

Black board example 7.4

A 0.500 kg mass is hung from a spring extending the spring by a distance x = 0.2 m

(a) What is the spring constant of the spring?

(b) How much work was done on the spring (by the spring force)?

(c) How much work was done by gravity on the spring?

A) -0.49J B) 0J C) 0.49J D) 0.98J E) 24.5J

Work-Kinetic Energy Theorem

WKKK if

xFmvmv

xavv

amF

xif

xif

xxnet

22

22

,

2

1

2

1

2

Change in the kinetic energy of a particle = net work done on the particle.

The kinetic energy of a particle is: 21

2K m v

A cannon ball of mass m = 1 kg moves at 500 m/s.

A truck of mass m = 10,000 kg moves at 5 m/s

Which has more kinetic energy?

A. The truck

B. The cannon ball

C. Same

D. Need more information

Black board example 7.5

Work due to friction

If friction is involved in moving objects, work has to be done against the kinetic frictional force.

This work is:dfW kf

Black board example 7.6

Angus is pulling a 10,000 kg truck with all his might (2000N) on a frictionless surface for 10.0 m.

(a) How much work is the man doing?

(b) What is the speed of the truck after 10 m.

(c) What is the speed of the truck after 10 m if there is friction? (friction coefficient: 0.0153)

A man loads a refrigerator onto a truck using a ramp. He thinks about the Physics of lifting it straight up versus rolling it up the ramp. (Ignore friction).

Which requires

a larger force? a longer distance? more work?

A. Lifting. A. Lifting A. Lifting

B. Same. B. Same B. Same

C. Rolling C. Rolling C. Rolling

Black board example 7.7

Power

Power is the rate at which work is done: dt

dWP

Average power (work done per time interval Dt):t

WP

vFdt

sdF

dt

dWP

The power can also be expressed as:

Dot product

The units of power are joule/sec (J/s) = Watt (W)James Watt (1736-1819); Scottish inventor and engineer whose improvements to the steam engine were fundamental to the changes wrought by the Industrial Revolution.(from Wikipedia)

Black board example 7.8

An elevator having a total mass of 1800 kg moves upward against a frictional force of 4000N at a constant speed of 3 m/s.

(a) What is the power delivered by the motor?