Physics Chapter 5

Work formula

F x d

• f = force

• d = displacement

The work, W, done by a constant force during a linear displacement along the x-axis is

W = F_x Δx

• F_x = x-component of the force

• Δx = objects displacement

The work equation applies when

the force is in the same direction as the displacement

W = (F cos θ) Δx

• F = the magnitude of the force

• Δ x = the magnitude of the object’s displacement

• θ = angle between F and Δx

work gives no information about

• The time it took for the displacement to occur
• The velocity or acceleration of the objec

work is a ____ quantity

scalar

• no direction associated with it

Si unit of Work

newton x meter = Joule

Joule is equal to

kg x m² / s²

The work done by a force is zero when

the force is perpendicular to the displacement

• cos 90^o = 0

If there are multiple forces acting on an object, the total work done is

the algebraic sum of the amount of work done by each force

The total work must be found for x, y, and z directions

separately

when work is zero

• displacement is horizontal

• force is vertical

• cos90= 0 since work, W = (Fcos theta)d

• W= 0 x J

work can be

positive or negative

positive work

gained energy

• the force and the displacement are in the same direction

negative work

lost energy

• the force and the displacement are in the opposite direction

Positive and negative work done simply provide

energy exchange information

work being negative or positive example

Kinetic energy

Energy associated with the motion of an object of mass m moving with a velocity v

• KE = ½ mv²

mass (m) in kinetic energy formula is a ____ quantity

scalar

velocity (v) in kinetic energy formula is a ____ quantity

vector

• not related to speed (which is scalar)

KE in kinetic energy formula is a ____ quantity

scalar (same units as work)

work is related to

kinetic energy

Work-Kinetic Energy Theorem (WET)

When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy

• W_net = KE_f - KE_i = ΔKE

in W_net = KE_f - KE_i = ΔKE, speed will increase if

net work is positive

in W_net = KE_f - KE_i = ΔKE, speed will decrease if

net work is negative

Potential energy

associated with the position of the object within some system

• property of the system, not the object

system

a collection of objects interacting via forces or processes that are internal to the system

Gravitational Potential Energy

the energy associated with the relative position of an object in space near the Earth’s surface

• objects interact with the earth through the gravitational force

• the potential energy is for the earth-object system

Work and Gravitational Potential Energy

• PE = mgy = mgh
• W_gravity = -mg(y_f - y_i )
• Units of Potential Energy are the same as those of Work and Kinetic Energy

• Joule (J)

• GPE/PEg=mgh

• h=y

To consider non-conservative forces and gravity, the work-energy theorem can be extended to include potential energy

If other conservative forces are present in the work energy theorem, potential energy functions can be developed for them and their change in that potential energy added to the right side of the equation

ΔPE_s = spring potential energy

ΔPE_g if spring is horizontal

not present/taken out of equation

There are two general kinds of forces

1. conservative

2. nonconservative

conservative force

Work and energy associated with the force can be recovered

Nonconservative force

The forces are generally dissipative, and work done against it cannot easily be recovered

A force is conservative if the work it does on an object moving between two points is

independent of the path the objects take between the points

• The work depends only upon the initial and final positions of the object
• Any conservative force can have a potential energy function associated with it

Examples of conservative forces include:

• Gravity
• Spring force
• Electromagnetic forces

Potential energy is another way of looking at the work done by

conservative forces

A force is nonconservative if the work it does on an object depends on

the path taken by the object between its final and starting points

Examples of nonconservative forces

Kinetic friction (see next slide), air drag, propulsive forces

Friction Depends on the Path

• The blue path is shorter than the red path
• The work required is less on the blue path than on the red path
• Friction depends on the path and so is a non-conservative force

work-energy theorem can be expressed in terms of the work done by both

conservative forces, W_c, and nonconservative forces, W_nc

To say a physical quantity is conserved is to say that

the numerical value of the quantity remains constant throughout any physical process, although the quantities may change its form

In Conservation of Energy, the total mechanical energy remains

constant

In any isolated system of objects interacting only through conservative forces, the total mechanical energy of the system remains

constant

Law of Conservation of Energy (LCE)

we can neither create nor destroy energy

• another way of saying energy is conserved

• if the total energy of a system does not remain constant, the energy must have crossed the boundary by some mechanism

• system needs to be defined precisely

Total mechanical energy is the sum of the

kinetic and potential energies in the system

• Other types of potential energy functions can be added to modify this equation

• ΔE = 0 because no energy change (same throughout)

A location where the gravitational potential energy is zero must be chosen for

each problem

• The choice is arbitrary since the change in the potential energy is the important quantity
• Once the position is chosen, it must remain fixed for the entire problem
• Choose a convenient location for the zero-reference height
• Often the Earth’s surface
• May be some other point suggested by the problem

Potential Energy Stored in a Spring

The force used in stretching or compressing a spring is a conservative force

• involves spring constant, k

• displacement is directly proportional to the applied force

• Hooke’s Law gives the force

Hooke’s Law

F_s = - kx

• F_s is the restoring force

• F_s is in the opposite direction of x

• k depends on how the spring was formed, the material it is made from, thickness of the wire, etc.

• negative sign depends on direction