# PHY C6: Work & Energy

Today we’ll be covering:

• Energy
• Definition
• Types of energy
• Conservation of Energy
• Work
• Work in terms of Force & Displacement
• Work done by an expanding Gas
• Efficiency

Let’s get to it!

What is Work?
Work is done when a force moves an object in the direction of the force.
The magnitude is a PRODUCT of FORCE & DISPLACEMENT along the DIRECTION of force.

If the FORCE (F) & the DISPLACEMENT (d) are in the same direction:
Work done = Fd

If they are in DIFFERENT directions:
Work done = Fd cos θ

 If the object travels in the direction of force, the work done is POSITIVE. Work is said to be done BY the force. For example, a pulling force moves a box 3 m to the right. Work is done BY the pulling force, ON the box. If the object travels in the OPPOSING direction of force, the work done is NEGATIVE. Work is said to be done ON or AGAINST the force. For example, a box experiences frictional force OPPOSITE the direction of motion Work is done AGAINST the friction, BY another force (pushing force). If the object moves PERPENDICULAR to the direction of force, NO WORK is done (because the perpendicular component is 0). For example, a person lifts up a box & walks forward. No work is done on the box.

Work is expressed in Newton-Metres (Nm) OR Joules (J)
Since work is equivalent to energy, they both share the same unit: Joules (J).

Work is a SCALAR, since it is a product of 2 vectors.

How do you deal with multiple forces?
If multiple forces are acting on the object, each force does different magnitudes of work.
The NET value of the work done ON the object is the product of NET force & displacement.

For example, a box being pushed by a person on a rough surface from rest to a constant speed, across a distance.

• The work done by the person is the force he exerts x distance: it has a POSITIVE value.
• The work done by the frictional force has a NEGATIVE value (since friction always acts OPPOSING direction of motion).
• The work done ON the box is the NET force x distance travelled.
W = Fpersond – Ffrictiond
W = (Fperson – Ffriction)d

In terms of energy (which we will explore below),
The energy used by the person is converted to kinetic energy & heat due to friction.

What is Work Done by a Gas?
The product of the gas pressure & change in volume of the gas.
Work = pΔV

When a gas expands (increases in volume) & maintains a CONSTANT pressure, WORK is done by the gas.

For example, take a gas trapped in a cylinder with a movable piston.
The gas exerts a constant pressure (p) on the walls of the piston.

When the gas expands (due to a change in temperature or any other energy input),
the force of collisions against the piston (F) push the piston across a distance (s).

Since pressure = Force/Area
p = F/A
F = pA
Work = pAs

Since As = change in volume of gas
As = ΔV
Work = pΔV

 When the gas volume INCREASES, POSITIVE work is done BY the gas. When the gas volume DECREASES. NEGATIVE work is done BY the gas OR POSITIVE work is done ON the gas

If it is said that the gas within a container is in EQUILIBRIUM with the external pressure,
you can conclude that the pressure of the gas is EQUAL to the external pressure.

For example:
A gas in a container expands by 2 m3 against an external pressure of 100 kPa.
Work done by the gas is 2 m3 x 100000 Pa = 200 kJ

What is Energy?
The ability to do work.
The more energy possessed by a body, the more work can be done.
Energy comes in many forms, & can be converted between these forms.

Energy is measured in Joules (J) (Newton-Metres).
It is a SCALAR just like work.

You can also measure energy in other units, which we’ll come across later on:

• kilowatt-hours (kWh), 1 kWh = 3.6 x 106 J
• electron-volts (eV), 1 eV = 1.6 x 10-19 J

A few Types of Energy:

 Energy Definition Kinetic Energy Energy due to motion Gravitational Potential Energy Energy due to position of a mass in a grav. field Elastic Potential Energy Energy due to stretching/compressing of an object Electrostatic Potential Energy Energy due to position of a charge in an electric field Electrical Energy Energy due to moving charges Electromagnetic Radiation Energy associated with EM waves Thermal Energy Energy due to internal kinetic energy of particles (heat) Chemical Energy Energy stored in chemical bonds Nuclear Energy Energy stored in the nuclei of atoms Internal Energy Total energy within an object: includes thermal, chemical, & nuclear energy Sound Energy Energy transferred from particle to particle through sound waves

What is the Relationship between Energy & Work?
Work causes the TRANSFER of energy between objects.
A certain type of energy can be CONVERTED into other forms.

Work done = Amount of energy transferred
Fs = ΔE

For example,
A car burns some fuel & moves a distance (s) under a certain force applied by the engine (F), & releases some heat.
The amount of chemical energy transformed into kinetic energy + heat energy = Force x displacement.

Ekinetic + Eheat = Fs

What is the Law of Conservation of Energy?
“Energy cannot be created or destroyed, only converted into different forms.”

This means the TOTAL energy of a CLOSED SYSTEM is conserved (constant) at all points in time.
Most of the time, in our daily experience, a source of energy is converted into USEFUL energy output & WASTEFUL energy output.
The TOTAL amount of energy stays the same.

Einput = Euseful output + Ewasteful output

In the case of the car above,
the kinetic energy was USEFUL,
the heat energy was WASTEFUL.
Total energy input = total energy output
Echemical = Ekinetic + Eheat

Thus, if there were any unknown values, we could use the equation to find them.

What is Efficiency?
USEFUL output divided by the total input energy.

Efficiency = useful energy output/total energy input x 100

In the case of the car above,
the efficiency of the car = Ekinetic / (Ekinetic + Eheat) x 100
OR Ekinetic / Echemical x 100

Power can also be used instead of energy to calculate the efficiency – it would yield the same answer.