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### Work and Energy

Chapter 6

Expectations

After Chapter 6, students will:

- understand and apply the definition of work.
- solve problems involving kinetic and potential energy.
- use the work-energy theorem to analyze physical situations.
- distinguish between conservative and nonconservative forces.

Expectations

After Chapter 6, students will:

- perform calculations involving work, time, and power.
- understand and apply the principle of conservation of energy.
- be able to graphically represent the work done by a non-constant force.

The Work Done by a Force

The woman in the picture exerts a force F on her suitcase, while it is displaced through a distance s. The force makes an angle q with the displacement vector.

The Work Done by a Force

The work done by the woman is:

Work is a scalar quantity. Dimensions: force·length

SI units: N·m = joule (J)

History/Biography Break

James Prescott Joule

December 24, 1818 –

October 11, 1889

English physicist, son of a wealthy brewer, born near Manchester. He was the first scientist to propose a kinetic theory of heat.

The Work Done by a Force

Notice that the component of the force vector parallel to the displacement vector is F cos q. We could say that the work is done entirely by the force parallel to the displacement.

The Work Done by a Force

Recalling the definition of the scalar product of two vectors, we could also write a vector equation:

The Work Done by a Force

Work can be either positive or negative.

In both (b) and (c), the man is doing work.

(b): q = 0°;

(c): q = 180°;

A Force Accelerates an Object

Let’s look at what happens when a net force F acts on an object whose mass is m, starting from rest over a distance s.

A Force Accelerates an Object

The object accelerates according to Newton’s second law:

A Force Accelerates an Object

Applying the fourth kinematic equation:

Kinetic Energy

A closer look at that result:

We call the quantity kinetic energy.

In the equation we derived, it is equal to the work (Fs) done by the accelerating force.

Kinetic energy, like work, has the dimensions of force·length and SI units of joules.

The Work-Energy Theorem

The equation we derived is one form of the work-energy theorem. It states that the work done by a net force on an object is equal to the change in the object’s kinetic energy. More generally,

If the work is positive, the kinetic energy increases. Negative work decreases the kinetic energy.

The Work-Energy Theorem

A hand raises a book from height h0 to height hf, at

constant velocity.

Work done by the hand force, F:

Work done by the gravitational force:

The Work-Energy Theorem

Total (net) force exerted

on the book: zero.

Total (net) work done

on the book: zero.

Change in book’s kinetic energy:

zero.

The Work-Energy Theorem

Now, we let the book fall freely

from rest at height hfto height h0.

Net force on the book: mg.

Work done by the gravitational

force:

The Work-Energy Theorem

Calculate the book’s final kinetic

energy kinematically:

The book gained a kinetic energy equal

to the work done by the gravitational force

(per the work-energy theorem).

Gravitational Potential Energy

The quantity

is both work done on the

book and kinetic energy gained by

it. We call this the gravitational

potential energy of the book.

Work and the Gravitational Force

The total work done by the

gravitational force does not

depend on the path the book takes.

The work done by the gravitational

force is path-independent. It

depends only on the relative

heights of the starting and

ending points.

Work and the Gravitational Force

Over a closed path (starting and ending points the same), the total work done by the gravitational force is zero.

Forces and Work

Compare with the frictional force. The longer the path, the more work the frictional force does. This is true even if the starting and ending points are the same. Think about dragging a sled around a race course.

The work done by

the frictional force

is path-dependent.

Conservative Forces

The gravitational force is an example of a conservative force:

- The work it does is path-independent.
- A form of potential energy is associated with it (gravitational potential energy).
Other examples of conservative forces:

- The spring force
- The electrical force

Nonconservative Forces

The frictional force is an example of a nonconservative force:

- The work it does is path-dependent.
- No form of potential energy is associated with it.
Other examples of nonconservative forces:

- normal forces
- tension forces
- viscous forces

Total Mechanical Energy

A man lifts weights upward at a constant velocity.

He does positive work on the weights.

The gravitational force does equal negative work.

The net work done on

the weights is zero.

But …

Conservation of Mechanical Energy

The gravitational potential energy of the weights

increases:

The work done by the nonconservative normal force of

the man’s hands on

the bar changed the

total mechanical

energy of the weights:

Conservation of Mechanical Energy

Work done on an object by nonconservative forces

changes its total mechanical energy.

If no (net) work is done by nonconservative forces, the

total mechanical

energy remains

constant (is conserved):

Conservation of Mechanical Energy

This equation is another form of the work-energy theorem.

Note that it does not require both kinetic and potential energy to remain constant – only their sum. Work done by a conservative force often increases one while decreasing the other. Example: a freely-falling object.

Conservation of Every Kind of Energy

“Energy is neither created nor destroyed.”

Work done by conservative forces conserves total mechanical energy. Energy may be interchanged between kinetic and potential forms.

Work done by nonconservative forces still conserves total energy. It often converts mechanical energy into other forms – notably, heat, light, or noise.

Power

Power is defined as the

time rate of doing work.

Since power may not

be constant in time, we

define average power:

SI units: J/s = watt (W)

James Watt

1736 – 1819

Scottish engineer

Invented the first efficient

steam engine, having a

separate condenser for the

“used” steam.

Graphical Analysis of Work

Plot force vs. position (for a constant force):

Graphical Analysis of Work

Plot force vs. position (for a variable force):

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