Chapter 6
The Giancoli onLine Tutor

Work and Energy

I. Overview

Definition of terms:

Conservative Force: A Force which can be associated with a Potential Energy.
Non-Conservative Force: A Force, such as, friction or air resistance, which converts Mechanical Energy into other forms of energy, such as, heat.

Potential Energy:

Spring Potential Energy	kx²/2	x is the distance the spring is compressed or stretched. k is the spring constant, usually a given.
Gravitational Potential Energy	mgy	y is the height as measured from any reference level

Symbol Table:

Symbol	Name	Explanation	Formula
W	Work	The product of a force on an object, the distance it moves, and the cosine of the angle between the direction of motion and the direction of the force	W=FDcos(q)
W_T	Total Work	The work done on an object by all the forces acting on it.	W=F_TDcos(q)
W_CF		The work done by all the conservative forces acting on an object.
W_NCF		The work done by all the non-conservative forces acting on an object.
PE	Potential Energy		kx²/2 and/or mgy
KE	Kinetic Energy		mv²/2
ME	Mechanical Energy		KE+PE

Relationships:

W_T=KE₂-KE₁	The total work done on an object from time "1" to time "2".
W_CF=-(PE₂-PE₁)	The work done by a conservative force (or forces) equals the negative of its change in potential energy.
W_NCF= (ME₂-ME₁)	The work done by non-conservative forces equals the change in mechanical energy

II. Energy Problems Without Friction

Setting up the Problems:

The simplest problems are those that do not involve non-conservative forces (no friction etc.). Most problems will be of this nature. Since
W_NCF=0
it follows that
ME₂=ME₁.
which means that the KE+PE at any point equals the KE+PE at any other point.

(mv²/2+mgy and/or kx²/2)₂=
(mv²/2+mgy and/or kx²/2)₁

Run the movie a couple of times then use the buttons on the right to step back and forth through it. Notice that both the KE and the PE change, but the ME is always the same and is the sum of the KE and PE. (In this simulation the mass is assumed to be 1 kg and g= 10 m/s².)

If you don't have the quicktime plugin you can view a GIF animation version of the movie here.

TIPS and Strategies:

When working a problem of this type, pick the starting point as your first point. This is almost always true for pendulums, cars on hill, roller coasters, masses on springs etc. The second point will usually be the location of the object at the end of the problem statement.
Suppose that are asked to find the speed of a 1 kg pendulum bob (mass) at its lowest point if it is released (started from) an initial height of 1 meter. At the start (point 1) the mass is not moving so its KE=0. As a result its

ME₁= PE₁= mgy= 1 kg*10m/s2*1m= 10J

at the bottom of its swing y (the height) is zero and thus so is the PE. So

ME₂= KE₂= mv²/2= 1 kg*v²/2.

Setting ME₂=ME₁ we find:

from which we obtain v,
v₂= 20.
In problems like this the mass will cancel out and frequently will not be given. If it had not been given we would have found that:

m*10*1= m*v²/2,

The mass will cancel out and we have the same result as before.
Any force that is perpendicular to the motion of an object can not do work (cos90^o=0) and thus does not enter into any of these problems. This will always be true for the Normal Force and for the tension force in the string of a pendulum.

Interactive Example
What is the work done by the string in the above movie? Enter answer and click outside.
Remember, you can only work a problem by starting it first. SO START! If nothing else draw a picture.

Interactive Examples

A pendulum has a mass of 2-kg ,a length of 1.2 meter and swings through a (half) arc of 37 degrees. To the tenth of a Joule what is its maximum Kinetic Energy? Enter answer and click outside.

( Use 10 for g)
ANSWER:

A car traveling at 72 kmh at the bottom of a hill runs out of gas, ignoring friction and air resistance, how high up the next hill can it go?
( Use 10 for g)
Enter your answer below and click outside.
ANSWER:

III. Energy Problems with Friction

Problems with friction or air resistance (which we will treat the same way) are worked very similarly. If the question is to find the work done by friction use:

W_NCF=f=ME₂-ME₁

For example, If you were told that the speed of the pendulum bob at the bottom of its first swing, in the above example, was 4m/s and asked to determine the work done by air resistance you would set:

W_f= ME₂-ME₁= mv²/2-mgy=1*4²/2-1*10*1= -2J,

and that would be your answer.

If you want to find the average force of air resistance in the above example you can use the fact that the work done by friction (or friction like force) is always equal to -f*D.
(The minus sign comes from the cos(180^o)=-1... The force of friction is always opposite the motion)

If the arc length between the top (release point) and the bottom is 4m (and it is) that would mean the work done by the friction like force would equal -f*4, but we also know that it equals -2 J so:

-f*4=-2

or f=.5 N.

Interactive Example

Suppose the car in the previous Example has a mass of 2000 kg and only makes it up the hill to a height of 15 m, what is the work done by friction? Enter answer and click outside.

ANSWER:

Suppose the car in theExample above travels 100 m along the road before it comes to rest at the height of 15 m, assuming that the friction force is constant, what is it to the nearest newton? Enter answer and click outside.

ANSWER: