Calculus-Based Physics by Jeffrey W. Schnick - HTML preview

Chapter 24 Work and Energy

On the left we have the work W, so:

W = m a r

∆

On the right we have two quantities used to characterize the motion of a particle so we have

certainly met our goal of relating work to motion, but we can untangle things on the right a bit if

we recognize that, since we have a constant force, we must have a constant acceleration. This

means the constant acceleration equations apply, in particular, the one that (in terms of r rather

than x) reads:

v 2 = v 2

o

+ 2a∆r

Solving this for a∆r gives

1

1

2

2

a ∆r =

v − v

o

2

2

Substituting this into our expression for W above (the one that reads W = m a r

∆ ) we obtain

 1

1

2

2 

W = m  v −

vo 

 2

2



which can be written as

1

1

2

2

W =

v

m

−

v

m o

2

2

1

Of course we recognize the

2

v

m o as the kinetic energy of the particle before the work is done

2

1

on the particle and the

2

v

m

as the kinetic energy of the particle after the work is done on it.

2

To be consistent with the notation we used in our early discussion of the conservation of

mechanical energy we change to the notation in which the prime symbol ( ′ ) signifies “after” and

no super- or subscript at all (rather than the subscript “o”) represents “before.” Using this

notation and the definition of kinetic energy, our expression for W becomes:

W = K ′ − K

Since the “after” kinetic energy minus the “before” kinetic energy is just the change in kinetic

energy ∆K, we can write the expression for W as:

W = K

∆

(24-2)

This is indeed a simple relation between work and motion. The cause, work on a particle, on the

left, is exactly equal to the effect, a change in the kinetic energy of the particle. This result is so

important that we give it a name, it is the Work-Energy Relation. It also goes by the name: The

Work-Energy Principle. It works for extended rigid bodies as well. In the case of a rigid body

that rotates, it is the displacement of the point of application of the force, along the path of said

point of application, that is used (as the ∆r ) in calculating the work done on the object.

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