Section 6.4.1Work

AP CalculusMay 19, 2010Berkley High School, D1B1

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 2

First, “Force” in SI

Mass (m) is measured kilograms

the remainder is the second derivative of position acceleration measured in

m/s2

Force (F) is measured in Newtons (N)

2

2

d sF m

dt

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 3

Second, “Force” in English

Is measured in… pounds (lbs)

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 4

Third, “Work” in SI

Force (F) is measured in Newtons (N)

Distance (d) is measured in meters (m)

Work is measured in Joules (J)

W Fd

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 5

Fourth, “Work” in English

Force (F) is measured in pounds (lb)

Distance (d) is measured in feet (ft)

Work is measured in ft-lbs

W Fd

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 6

Example:

How much work is done in lifting a 1.2 kg book off the floor to put it on a desk that is .7 m high? Remember that acceleration due to gravity is 9.8 m/s2.

2

2

1.2 9.8

11.76 N

11.76 .7 8.232 J

d sF m

dtF

F

W Fd

W

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 7

Example:

How much work is done in lifting a .54 lb book off the floor to put it on a desk that is 2.3 ft high?

.54 2.3 1.242 ft-lb

W Fd

W

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 8

Example:

When a particle is located at a distance x feet from the origin, a force of x2+2x pounds acts on it.

How much work is done in moving it from x=1 to x=3?

2

short

3

total

1

32

total

1

33 2

total1

3 2 3 2total

total

( ) 2

( )

( )

2

1 12

3 2

1 13 3 1 1

3 3

4 509 9 ft lbs

3 3

F x x x

W F x dx

W F x dx

W x x dx

W x x

W

W

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 9

Hooke’s Law (work done on and by springs) Hooke’s Law is concerned with the amount of

force needed to hold a spring away from its natural length

The law states that the force varies directly to the displacement from natural position.

For example, at natural length, it requires no force.

It requires twice as much force to hold a spring two meters from natural length as it does to hold it 1 meter.

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 10

Hooke’s Law (SI)

F=kx where… F is force measured in…

Newtons (N) x is the displacement of the length of the spring

from its natural length measured in… meters (m)

k is the spring constant (which varies from spring to spring) measured in… kilograms per seconds2 (kg/s2)

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 11

Hooke’s Law (English)

F=kx where… F is force measured in…

pounds (lbs) x is the displacement of the length of the spring

from its natural length measured in… feet (ft)

k is the spring constant (which varies from spring to spring) measured in… pounds per foot (lbs/ft)

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 12

Example:

A spring requires 20 Newtons of force to hold it at 15 cm. Its natural length is 10 cm.

How much work is done in pulling it from 15 cm to 20 cm?

.10

.05

.102

.05

2 2

20 .05

400

400

1400

2

200 (.1 .05 )

200 (.01 .0025)

1.5 Joules

F kx

k

k

W xdx

W x

W

W

W

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 13

Example:

A spring requires 15 Joules of work to move it from 20 cm to 25. Its natural length is 10 cm.

How much work is done in pulling it from 15 cm to 20 cm?

.15

.10

.152

.10

2 2

15

115

2

30 .15 .10

30 .0225 .01

30 (.0125)

2400

kxdx

k x

k

k

k

k

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 14

Example:

A spring requires 15 Joules of force to move it from 20 cm to 25. Its natural length is 10 cm.

How much work is done in pulling it from 15 cm to 20 cm?

.10

.05

.102

.05

2 2

2400

12400

2

1200 (.1 .05 )

1200 (.01 .0025)

9 Joules

W xdx

W x

W

W

W

Calculus, Section 6.4, Todd Fadoir, CASA, 2004 15

Assignment

Section 6.4, 1-10 all