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6. Work, Energy, and Power

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6. Work, Energy, and Power

The Dot Product

The dot product is the scalar

where q is the angle between the vectors and A

and B are their magnitudes.

A few properties of the dot product:

The definition of the dot product is consistent

with standard trigonometric

relationships. For example:

Law of cosines

The definition

implies

where

Work and Kinetic Energy

So far we have solved motion problems by

1. adding up all the forces to get the net force2. and applying Newton’s laws, e.g.,

2nd Law

Another way to solve such problems is to use an

alternative form of Newton’s laws, based on

energy principles.

We are about to deduce an important energy

principle from Newton’s laws.

Today, physicists view energy principles,

such as the conservation of energy, as

fundamental laws of Nature that are independent

of the validity of Newton’s laws.

We start with the concept of kinetic energy.

First, take the dot product of the 2nd law with

the velocity v

Next, integrate both sides

with respect to time

along a path from

point A to point B

where

is the kinetic energy

When the right-hand side is integrated, we

obtain the difference between the final and

initial kinetic energies, K2 and K1, respectively:

The left-hand side

can be rewritten as

The quantity

is called net work

The net work, W, done by the net force on an

object equals the change, ΔK, in its kinetic energy.

Energy is measured in joules (J):

J=N m

Work can be positive or negative.

Kinetic energy is always positive.

Work-Kinetic Energy Theorem – Examples

A truck of mass 3000 kg is to

be loaded onto a ship using

a crane that exerts a force of

31 kN over a

displacement of 2m.

Find the upward speed of

truck after its displacement.

Two forces act on the truck:

1. Gravity w

2. Force of crane Fapp

Apply the work-kinetic energy

theorem

Since the forces are constant over

the displacement, we can write

the work as

that is, as the dot product of

the net force and the

displacement.

Work done on truck by gravity

Work done on truck by crane

From the work-kinetic energy theorem

we obtain:

Hook’s Law

Find work done on block

for a displacement, Δx = 5 cm

Find speed of block

at x = 0

m = 4 kg

k = 400 N/m

Compute work done

m = 4 kg

k = 400 N/m

m = 4 kg

k = 400 N/m

Now apply work-kinetic energy theorem

→

vi initial speed

vf final speed

m = 4 kg

k = 400 N/m

Why did we ignore gravity

and the normal force?

Speed at x = 0

m = 4 kg

k = 400 N/m

Power

Power is the rate at which work is done, or

energy produced, or used.

If the change in work is ΔW, in time interval

Δt, then the average power is given by

while the instantaneous power is

The SI unit of power is the watt (W) named

after the Scottish inventor James Watt.

W = J / s

Example: A 100 watt light bulb converts

electrical energy to light and heat at the rate

of 100 joules/s.

Given a force F and a small displacement dr

the work done is

therefore, the power can be written as

that is, the dot product of the force and the

velocity.

A cyclist who wants to move at velocity v

while overcoming a force F must produce a

power output of at least P = Fv. At 5 m/s

against an air resistance of F = 30 N, P = 150 W.

However, even going up a gentle slope of 5o, an

82 kg cyclist (+ bike) needs to output 500 W!

- The work-energy theorem relates the net work done on an object to the change in its kinetic energy: W = ∆K
- Work done on an object by a force is
- Poweris rate at which work is done