Chapter 6
Energy
Force is the agent of change; energy is a measure of change.
Work
Work is the change in the energy of a system resulting from the application
of a force acting over a distance.
This equation is for forces in the same direction as the motion. If the force is not in the same direction as the motion, you must find the force in the direction of motion, usually using cosine.
If the force that is used is gravity work is calculated using:
The units of Work are Nm which are joules, J.
Now we will do some examples.
Example 1
An
intern pushes a 72 kg patient on a 15 kg gurney, producing an acceleration
of 
.
How much work does the intern do by pushing the patient and gurney through
a distance of 2.5 m? Assume the gurney moves without friction.
Example 2
A 75.0 kg person slides a distance of 5.00 m on a straight water slide, dropping through a vertical height of 2.50 m. How much work does gravity do on the person?
Example 3
You
want to load a box into the back of a truck. One way is to lift it straight
up through a height h, as shown, doing a work 
.
Alternatively, you can slide the box up a loading ramp a distance L, doing
a work 
.
Assuming the box slides on the ramp without friction, which of the following
is correct: (a) 
,
(b) 
,
(c) 
?
Example 4
A
car of mass m coasts down a hill inclined at an angle 
below
the horizontal. The car is acted on by three forces: (i) the normal force 
exerted
by the road, (ii) a force due to air resistance, 
,
and (iii) the force of gravity, 
.
Find the total work done on the car as it travels a distance d along the
road.
Example 5
The
figure shows an overhead view of a puck on a frictionless horizontal surface.
Three constant horizontal forces act on the puck in the directions indicated.
The magnitude of 
is
10.0 N, that of 
is
15.0 N, and that of 
is
12.0 N. The puck starts from rest. What is the net work W done on the puck
by the three forces when the puck has gone through a displacement of magnitude
d = 0.400 m?
Mechanical Energy
Kinetic Energy is the energy of motion of any moving object.
The Work-Energy Theorem:
Example 6
A 4.1 kg box of books is lifted vertically from rest a distance of 1.6 m by an upward applied force of 60.0 N. Find (a) the work done by the applied force, (b) the work done by gravity, and (c) the final speed of the box.
Example 7
The
figure shows two industrial spies sliding an initially stationary 225 kg
floor safe a displacement
of
magnitude 8.50 m, straight toward their truck. The push of
Spy 001 is 12.0 N, directed at an angle of 
downward
from the horizontal: the pull of
Spy 002 is 10.0 N, directed at 
above
the horizontal. The magnitudes and directions of these forces do not change
as the safe moves, and the floor and safe make frictionless contact. (a)
What is the net work done on the safe by forces and during
the displacement ?
(b) During the displacement, what is the work 
done
on the safe by the gravitational force 
and
what is the work 
done
on the safe by the normal force 
from
the floor? (c) The safe is initially stationary. What is its speed 
at
the end of the 8.50 m displacement?
Example 8
A
boy exerts a force of 11.0 N at 
above
the horizontal on a 6.40 kg sled. Find the work done by the boy and the final
speed of the sled after it moves 2.00 m, assuming the sled starts with an
initial speed of 0.500 m/s and slides horizontally without friction.
Example 9
During
a storm, a crate of crepe is sliding across a slick, oily parking lot through
a displacement 
while
a steady wind pushes against the crate with a force 
as
shown in the figure. (a) How much work does this force from the wind do on
the crate during displacement? (b) If the crate has a kinetic energy of 10
J at the beginning of displacement ,
what is its kinetic energy at the end of ?
Example 10
An
initially stationary 15.0 kg crate of cheese wheels is pulled, via a cable,
a distance L = 5.70 m up a frictionless ramp, to a height h of 2.50 m, where
it stops. See the figure. (a) How much work is
done on the crate by the gravitational force during
the lift? (b) How much work 
is
done on the crate by the force 
from
the cable during the lift?
Example 11
An
elevator cab of mass m = 500 kg is descending with speed 
when
its supporting cable begins to slip, allowing it to fall with constant acceleration 
.
See the figure. (a) During the fall through a distance d = 12 m, what is
the work done
on the cab by the gravitational force ?
(b) During the 12 m fall, what is the work done
on the cab by the upward pull of
the elevator cable? (c) What is the net work W done on the cab during the
fall? (d) What is the cab's kinetic energy at the end of the 12 m fall?
Work Done by a Variable Force
So far we have used a constant force for our work calculations, but we can also use a force that varies over time to calculate work. To do this we need to use a little bit of calculus. For a one-dimensional variable force we need to look at the Force vs. Distance graph. We then have to find the area under the curve. An estimate could be found using rectangles and adding them up to find the area. The more rectangles the greater the accuracy. But to get the exact answer you need to find the integral.
This equation works for a one-dimensional variable force. If the force is two or three dimensional you must add appropriate integrals.
Example 12
A 5.0 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as shown in the figure. How much work is done by the force as the block moves from the origin to x = 8.0 m?
Example 13
Force 
,with
x in meters, acts on a particle, changing only the kinetic energy of the
particle. How much work is done on the particle as it moves from coordinates
(2 m, 3 m) to (3 m, 0 m)? Does the speed of the particle increase, decrease,
or remain the same?
Example 14
What
work is done by a force 
,with
x in meters, that moves a particle from a position 
to
a position 
?