Energy
{
Work and Power
In physics, work has a very specific
definition.
 Work is done when:
 A force is exerted on an object
 An object has a displacement in
the same direction as the force
W=Fdcosq
 SI unit for work is Joule (J)
 1 J = 1N*m (=0.7376 ft*lb)

Work
A 15 kg block is dragged over a rough,
horizontal surface by a 70 N force acting at
20 degrees above the horizontal. The block
is displaced 5 m, and the coefficient of
kinetic friction is 0.3. Find the work done
by:
 The 70 N force
 The normal force
 The force of gravity
 The friction force
Example 1

We are only interested in the
component of Force that acts in the
same direction (along the same axis)
as the displacement.

By using the equation: W=Fdcosq, we
are finding the scalar product of the
two vectors, force and displacement.

The scalar product is also called the
dot product.
𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 = 𝐹 ∙ 𝑑
Work
𝐴 ∙ 𝐵 = 𝐴𝐵𝑐𝑜𝑠𝜃
The dot product is positive for angles
between 0 and 90, negative if the angle is
between 90 and 180 and zero if the vectors
are perpendicular.
Dot product
Start with the scalar products of the unit
vectors:
𝑖 ∙ 𝑖 = 𝑗 ∙ 𝑗 = 𝑘 ∙ 𝑘 = 1 1 𝑐𝑜𝑠0 = 1
𝑖 ∙ 𝑗 = 𝑖 ∙ 𝑘 = 𝑗 ∙ 𝑘 = 1 1 𝑐𝑜𝑠90 = 0
Calculating Dot Product
𝐴 ∙ 𝐵 = (𝐴𝑥 𝑖 + 𝐴𝑦 𝑗 + 𝐴𝑧 𝑘) ∙ (𝐵𝑥 𝑖 + 𝐵𝑦 𝑗 + 𝐵𝑧 𝑘)
= 𝐴𝑥 𝑖 ∙ 𝐵𝑥 𝑖 + 𝐴𝑥 𝑖 ∙ 𝐵𝑦 𝑗 + 𝐴𝑥 𝑖 ∙ 𝐵𝑧 𝑘
+𝐴𝑦 𝑗 ∙ 𝐵𝑥 𝑖 + 𝐴𝑦 𝑗 ∙ 𝐵𝑦 𝑗 + 𝐴𝑦 𝑗 ∙ 𝐵𝑧 𝑘
+𝐴𝑧 𝑘 ∙ 𝐵𝑥 𝑖 + 𝐴𝑧 𝑘 ∙ 𝐵𝑦 𝑗 + 𝐴𝑧 𝑘 ∙ 𝐵𝑧 𝑘
𝐴 ∙ 𝐵 = 𝐴𝑥 𝑖 ∙ 𝐵𝑥 𝑖 + 𝐴𝑦 𝑗 ∙ 𝐵𝑦 𝑗 + 𝐴𝑧 𝑘 ∙ 𝐵𝑧 𝑘
𝑨 ∙ 𝑩 = 𝑨𝒙 𝑩𝒙 + 𝑨𝒚 𝑩𝒚 + 𝑨𝒛 𝑩𝒛
Calculating Dot Product
A particle moving in the xy plane
undergoes a displacement
d=(2.0i+3.0j)m as a constant force
F=(5.0i+2.0j)N acts on the particle.
Calculate the work done by the force.
 Calculate the angle between the force
and the displacement.

Example 2
Energy is defined as the ability to do
work.
Energy dealing with motion and
forces is called mechanical:
Kinetic
Potential
Energy
Kinetic Energy is the energy of
motion.
1
𝐾 = 𝑚𝑣 2
2
Potential Energy is stored energy.
We will primarily use gravitational
potential energy.
𝑈 = 𝑚𝑔ℎ
Kinetic and Potential
Energy in a closed system cannot be
gained or lost; it is transferred from one
type to another.
𝐸𝑏𝑒𝑓𝑜𝑟𝑒 ± 𝑊𝑁𝐶 = 𝐸𝑎𝑓𝑡𝑒𝑟
Non-conservative forces (or dissipative
forces) can add or subtract to the
mechanical energy of the system.
Conservation of Energy
If the force changes over time, we need to look at
the force in small pieces to get a more accurate
picture of the work that is done.
W≅
𝑖 𝐹𝑖 𝑑𝑖
As we make the divisions smaller and smaller, our
picture of the work being done gets more accurate.
Which leads us to…
𝑊=
Varying Force
𝐹 𝑥 𝑑𝑥
Spring force is linearly related to amount
the spring is displaced (stretched or
compressed).
𝐹 = −𝑘𝑥
The spring force acts in the opposite
direction of the displacement.
Spring Forces
The spring constant varies from
spring to spring. It is dependent
on the spring’s material, shape
and size.
𝐹
𝑥
𝑘 = (Newton/meter)
Spring Constant
𝑥1
𝑊=
−𝑘𝑥 𝑑𝑥
𝑥0
𝑥1
= −𝑘
𝑥𝑑𝑥
𝑥0
1 2
= − 𝑘𝑥
2
(where x0 = 0 m)
Work on/by a Spring
Power is the rate at which work
is done.
𝑊
𝑃=
𝑡
…..measured in Watts
Power