Work
Colin Murphy, Kevin Su, and Vaishnavi Rao
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Work (J if force is in N, ft-lb if force is in lb)
Work = Force * distance
Force (N)= mass * acceleration
NOTE: If you are using feet and lbs, the lbs is a
measurement of force, but if the question
uses meters and kg, you must use the formula
for force above to convert it to acceleration
(usually 9.8 from gravity)
Example 1
a) Calculate the work required to move a 1.5kg
object 0.7 meters up.
First, F=m*acceleration, so F = 1.5kg*9.8=14.7
Then, W = F*d, so W= 14.7 * .7 = 10.29J
b) Calculate the work required to move a 1.5lb
object 7 ft up.
W = F*d, so W= 1.5 * 7 = 10.5ft-lb
• The work problems that you are going to
actually see have amounts of force that are
not necessarily constant, so you must use the
formula:
dx
• Where f(x) is the force at distance x, and the
object is being moved from point a to point b
Example 2
Hooke’s law states that the force required to
maintain a spring stretching x units beyond its
natural length is f(x)=kx, where k is a constant,
and x is the distance that the spring is displaced
from its natural length.
So, if 30N of force is required to stretch a spring
1.5m from its natural length of .5M, how much
work is required to stretch it from 2m to 3m?
First, 1.5*k = 30, so k=20
Then just integrate f(x)=kx.
dx= 60J
Example 3
Mr. Shay is relaxing while using a pulley to pull
one of his many cruise ships up a waterfall.
The ship is 150ft from the top of the waterfall,
the cord weighs 600lbs, and the ship weighs
107 lb. How much work is required to get the
ship up the waterfall?
As more of the cord is pulled up the cliff, there is less weight of cord to pull, so, if we
assume that 1ft of rope weighs 4lbs, since we assume that each section of rope is an
equal weight.
So, F = 4x + 107, with x being the distance from the top of the cliff.
We integrate this on the interval 0 to 150, and we get:
Example 4
Mr. Shay is pumping gruel (with a density of 1200kg/m3) for the
freshman who row his cruise ship out of a container with the
following dimensions:
1m
4m
5m
x
10m
How much work does it take to pump all of the gruel out of the
spout at the top of the vat?
Since to find force we use F=mass*acceleration, and use the acceleration of gravity to
First,find
youthat
need
calculate
thetoforce
slice
of the gruel
thetoforce
required
moverequired
one layerto
to move
the topone
of the
container
is
to9600(5-x)kg
the top of*9.8,
the tank.
which is equal to 94080(5-x)dx N.
We
can
now
multiply
the volume that we have by the density to
We represent the distance from the top of the tank as x, and using similar
find
theraise
mass,
which
shows us that the density of a slice is
triangles
find
the
following:
The
work to we
a single
layer to the top is F*d, and since the distance from the slice
9600(5-x)dxkg.
to the top of the spout is x+1, the work required to raise a single slice to the top of
the container is 94080(5-x)(x+1)dx.
We then multiply this value by the length of the container and the height of a
Finally, we integrate it to get:
slice (dx) to get the volume of the slice, which ends up being (40-8x)dx.
Worksheet Questions
1. A crane is lifting objects up in a construction
site.
JJ
a) How much work is done when a crane pulls a
load with a mass of 140 kilograms to a height of
50 meters?
b) If the same load is at 50 meters and is further
pulled to a height of 75 meters, how much work
is done?
Worksheet Questions
2. A particle located on the x-axis is moved along
by a force measured by. If the particle is
moved from the origin to a distance of 9, how
much work is done?
Worksheet Questions
3. 6 joules of work are done when a spring is
stretched from its natural length of 10
centimeters to 15 centimeters.
a) What is the spring constant
of this spring?
J
b) How much work is done when the spring is
stretched from 15 to 20 centimeters?
J
J
Worksheet Questions
4. A 300 pound cable is 100 feet long and hangs
vertically from the top of a tall building. How
much work is required to lift the entire cable
to the top? Leave your answer in feet-lb.
feet pounds
Worksheet Questions
5. A sphere with a radius of 5 is filled with water.
If the spout is 1 meter long, how much work is
required to pump all the water out? The
density of water is 1000kg/.