Lecture
10: Work and Kinetic Energy
-
ENERGY = a scalar quantity which is associated with the state
of one or more objects.
-
STATE = the condition of an object, example, a state of motion
or a state of rest.
-
KINETIC ENERGY (K) = the energy associated with the STATE
of motion of an object.
-
K = ½ mv2 .
-
PROPERTIES OF KINETIC ENERGY:
-
Kinetic energy can never be negative since mass is positive and v2
is positive.
-
It is a scalar quantity.
-
SI units: 1 Joule (J) = 1 N-m
A. WORK
1) Work is the transfer of energy through a force.
2) Example:
-
If I cause an object to accelerate by applying a force to the object, I
essentially increase the kinetic energy of the object. The force I applied
transferred energy to the system causing an increase in the kinetic energy.
Therefore, the force did positive work on the system
-
If I cause an object to slow down by applying a force to the object, I
essentially decrease the kinetic energy of the system. The system transferred
energy to me through the force and negative work was done.
3) FORMAL DEFINITION: Work is transferred to or from an object by
a force acting on an object. Energy transferred to the object is positive
work done, and energy transferred from an object is equal to negative work
done.
4) Work is a scalar quantity with SI units: 1 Joule (J) = 1 N-m.
B. WORK AND KINETIC ENERGY
1) Suppose I apply a net constant force to a particle in the horizontal
direction causing it to accelerate from some initial velocity to some final
velocity. How much work was done on the system?
2) We assumed that the force was constant and F = ma, therefore the
acceleration is constant.
3) The last equation is the work-energy theorem for kinetic energy.
Basically, it means that the work done by any force which causes an object
to change its state of motion is just equal to the change in the objects
kinetic energy.
4) What if the force is not constant, does the equation still apply?
The formula applies to both constant and variable forces.
C. WORK AND CONSTANT FORCES
1) If the force is constant, then the following equations for finding
work apply to that particular situation.
2) The choice of which equation to use depends on how the problem is set
up. In the second equation, the angle q, is
the angle between the force and the direction of motion. Therefore, you
must establish the direction of motion in order to solve problems correctly.
D. WORK AND VARIABLE FORCES
1) The graphs below represent the force versus displacement of a constant
force (A), and a variable force (B).
2) As you can see from the graph of the constant force, the area under
the curve is just the work done by the force. If the force is variable,
it is not that straightforward to calculate.
3) The solution is to divide the graph into small rectangles and attempt
to sum up the areas of each individual rectangle as in the figure below.
E. EXAMPLE OF A VARIABLE FORCE: THE SPRING FORCE
1) Consider the following diagrams involving a mass attached to a spring:
2) The spring can be in equilibrium, compressed or stretched state.
The spring force is also called the restoring force because this force
always wants to bring the spring back to its equilibrium position (see
figure above).
3) The force is given by Hooke’s Law: F = -kx
4) The negative sign indicates that the force is in a direction opposite
that of the displacement. If you displace it to the right, the force is
to the left and vice versa.
5) The constant k = spring constant or force constant which is a measure
of the stiffness of the spring. It is a scalar quantity with SI units of
N/m.
6) What is the work done by the spring force?
