WORK AND ENERGY

Welcome back to PHY 183

Meaning of the picture ? PE  KE
3.1 Work
3.2 Energy
3.3 Conservative and
nonconservative forces
3.4 Power, energy and
momentum conservation
3.5 Linear momentum
3.6 Collisions
3.7 Motion in a gravitational potential
CHAPTER 3 WORK AND ENERGY
Part 1
Work
Work of constant Force
Work done by sum of constant Forces
Work done by Variable Force




Review: Work of constant Force
Work ( W) of a constant force F acting through a displacement r is:

W = F r = F r cos() = Fr r (N.m=J)

Learning check
Find work done by tension T and
by weigh


Find work done by Normal force N.


What is your conclusion from these ??
WT= 0 and WW = ?
WN= 0
Test: Work done by gravity
WG =F r =mg. r.cos
= -mg y
(remember y = yf - yi)
WG = -mg y


Depends only on y !

Let mass m move on the path r by gravity only
Compute work done ??
Test: Work done by gravity
Let mass m move on the path (r1 + r 2+ . . .+ rn) by gravity
Compute work done on total path:
WNET = W1 + W2 + . . .+ Wn
Note: Force = weight= F = mg
= F r 1+ F r2 + . . . + F rn
= F (r1 + r 2+ . . .+ rn)
= F r
= F y


Depends only on y,
not on path taken!
Work done by sum of constant Forces
Suppose FNET = F1 + F2 + F3 …+ Fn and the displacement is r.



The work done by each force is:
W1 = F1 r W2 = F2 r
…. Wn = Fn r
Work done by Variable Force:
When the force was constant, we
wrote W = F x
area under F vs. x plot:


For variable force, we find the area
by integrating:
dW = F(x) dx.
Work of variable force
Example: Spring
For a spring we know that Fx = -kx.
Problem
Compute work done by the spring Ws during a displacement from x1 to x2 (Note: the F(x) vs x plot between x1 and x2).
Solution
Work by variable force in 3D
Work dWF of a force F acting
through an infinitesimal
displacement r is:
dW = F.r
The work of a big displacement through a variable force will
be the integral of a set of infinitesimal displacements:


WTOT = F.r
ò
Part 2
Energy
Kinetic energy KE: moving energy
Potential energy PE: Tendency for work
Total energy: TE=KE + PE


Kinetic Energy Theorem
{Net Work done on object}=
={change in kinetic energy of object}
 WF = K = 1/2mv22 - 1/2mv12
Prove: Kinetic Energy Theorem
for a variable Force
Test: Falling Objects
Three objects of mass m begin at height h with velocity 0. One falls straight down, one slides down a frictionless inclined plane, and one swings on the end of a pendulum. What is the relationship between their velocities when they have fallen to height 0?
Solution
Learning check
A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest.
If the initial speed of the box were doubled and its mass were halved, how far x2 would the spring compress ?
x1
Solution
Again, use the fact that WNET = DK.

In this case, WNET = WSPRING = -1/2 kx2
and K = -1/2 mv2

Problem: Spring pulls on mass.
A spring (constant k) is stretched a distance d, and a mass m is hooked to its end. The mass is released (from rest). What is the speed of the mass when it returns to the relaxed position if it slides without friction?
Step-1
First find the net work done on the mass during the motion from x = d to x = 0 (only due to the spring):





Step-2
Now find the change in kinetic energy of the mass:
Step-3
Now use work kinetic-energy theorem: Wnet = WS = K.
Step-4
Now suppose there is a coefficient of friction  between the block and the floor
The total work done on the block is now the sum of the work done by the spring WS (same as before) and the work done by friction Wf.
Wf = f.Δr = - mg d
Step-4
Again use Wnet = WS + Wf = K

Wf = -mg d
Part 3
Conservative and nonconservative forces

Conservative Forces:
In general, if the work done does not depend on the path taken (only depends the initial and final distances between objects), the force involved is said to be conservative.
Gravity is a conservative force:


Gravity near the Earth’s surface:


A spring produces a conservative force:
Result of conservative Forces
We have seen that the work done by a conservative force does not depend on the path taken.
W1
W2
W1
W2
W1 = W2
WNET = W1 - W2
= W1 - W1 = 0
Therefore the work done in a closed path is zero.
Potential Energy
For any conservative force F we can define a
potential energy function U in the following way:

The work done by a
conservative force is equal
and opposite to the change
in the potential energy function.
This can be written as:
Test: Gravitational Potential Energy
We have seen that the work done by gravity near the Earth’s surface when an object of mass m is lifted a distance y is
Wg = -mg y
Compute the change in potential energy of this object ??
U = -Wg = mg y
Gravitational Potential Energy
So we see that the change in U near the Earth’s surface is:
U = -Wg = mg y = mg(y2 -y1).
So U = mg y + U0 where U0 is an arbitrary constant.
Having an arbitrary constant U0 is equivalent to saying that we can choose the y location where
U = 0 to be anywhere we want to.