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6. Work, Energy, and Power. The Dot Product. The Dot Product. The dot product is the scalar. where q is the angle between the vectors and A and B are their magnitudes. . The Dot Product. A few properties of the dot product:. The Dot Product.

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The dot product1
The Dot Product

The dot product is the scalar

where q is the angle between the vectors and A

and B are their magnitudes.

The dot product2
The Dot Product

A few properties of the dot product:

The dot product3
The Dot Product

The definition of the dot product is consistent

with standard trigonometric

relationships. For example:

Law of cosines

The dot product4
The Dot Product

The definition



Energy principles
Energy Principles

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.

Energy principles1
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.

Kinetic energy
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


is the kinetic energy

Kinetic energy1
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 work kinetic energy theorem
The Work-Kinetic Energy Theorem

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.

Example lifting a truck
Example – Lifting a truck

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.

Example 2
Example (2)

Two forces act on the truck:

1. Gravity w

2. Force of crane Fapp

Apply the work-kinetic energy


Example 3
Example (3)

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


Example 4
Example (4)

Work done on truck by gravity

Work done on truck by crane

Example 5
Example (5)

From the work-kinetic energy theorem

we obtain:

Example compressed spring
Example – Compressed Spring

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

Example 21
Example (2)

Compute work done

m = 4 kg

k = 400 N/m

Example 31
Example (3)

m = 4 kg

k = 400 N/m

Example 41
Example (4)

Now apply work-kinetic energy theorem

vi initial speed

vf final speed

m = 4 kg

k = 400 N/m

Example 51
Example (5)

Why did we ignore gravity

and the normal force?

Speed at x = 0

m = 4 kg

k = 400 N/m


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


Example bicycling
Example – Bicycling

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