Work and Energy CHAPTER 6. A New Perspective on Motion We have been analyzing motion through the perspective of Newton’s Laws dealing with acceleration,

Work and EnergyCHAPTER 6

A New Perspective on Motion

We have been analyzing motion through the perspective of Newton’s Laws dealing with acceleration, velocity, and displacement.

In this chapter, we will look at motion from a different perspective – that of work and energy.

What is Work?

When a force acts upon an object and causes its displacement, it is said that work has been done on the object.

The force MUST cause a displacement! No work is done if the object doesn’t move.

Is There Work?

1. A woman pushes a shopping cart across the floor. YES – Force causing displacement.

2. A man pushes against a wall. NO – No displacement

3. A book falls off a table to the ground. YES – Gravity provides the force to cause displacement

4. A child holds a book over her head while she stands on a moving walkway. NO – The force is not causing the displacement.

Calculating Work

where F=force, d=displacement, and θ is the angle between the force and displacement vectors. What if the force and the displacement are in exactly the

same direction? Then , so . What if the force and displacement are in opposite directions? Then , so .

Calculating Work

where F=force, d=displacement, and θ is the angle between the force and displacement vectors. What if the force and the displacement vectors are

perpendicular? Then , so .

Units for Work

The SI unit for work is called the joule (J).

The English unit for work is foot-pound (ft-lb).

Example #1

A 100 N force is applied to move a 15 kg object a horizontal distance of 5 meters at a constant speed.

How much work must the friction force do to the object?

Example #2

A 100 N force is applied at an angle of 30° to the horizontal to move a 15 kg object a horizontal distance of 5 meters at a constant speed.

Example #3

An upward force is applied to lift a 15 kg object to a height of 5 meters at a constant speed.

Since a=0, we know

Gravity does -735 J of work!

Example #4

A mother carries a 5 kg stroller up 4 flights of stairs (for a vertical distance of 15 meters). She then pushes the stroller with a force of 25 N with a constant speed and at an angle of 35° from the horizontal for a horizontal distance of 3.5 meters. How much work does the mother do on the stroller during this entire process?

Vertical: Horizontal: Total:

Energy

Energy (E) – the ability to do work. Types of energy:

Mechanical – kinetic energy + potential energy Electrical Nuclear Heat Chemical Sound

Kinetic Energy

Kinetic energy is the energy of motion. Many types (like vibrational and rotational) For now we are talking about translational kinetic

energy – the energy due to motion from one location to another.

Measured in Joules (like work)

Kinetic Energy Example

Which object has greater kinetic energy – a 12 kg object moving at 20 m/s or a 24 kg object moving at 10 m/s?

Note that KE is directly proportional to , so for example as v triples, KE goes up by a factor of 9.

Work-Energy Theorem

When work is done on an object, the result is a change in the object’s kinetic energy.

or

This theorem deals with the work done by a net external force (not an individual force, unless it is the only force).

Work-Energy Theorem Example 1

Calculate the work required to take a 800 kg car from rest to 100 m/s.

“from rest” means that the second part goes away.

Work-Energy Theorem Example 2

A 6.4 kg object is pushed by a force of 15 N, resulting in a displacement of 12 m in the same direction as the push. If the object’s final speed is 10 m/s, find the initial speed of the object.

use

Work Done by the Force of Gravity

Recall that as an object falls, gravity is doing work on the object.

Since the force and displacement are the same direction (θ=0°) and the force is the weight of the object, the work equation becomes:

is the downward displacement of the object (how far it has fallen)

Work Done by Gravity Example 1

A 65 kg object falls from a height of 200 m to a height of 50 m. Calculate the work done by gravity.

Work Done by Gravity Example 2

A 65-kg gymnast springs vertically upward from a trampoline. The gymnast leaves the trampoline at a height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. Ignoring air resistance, determine the initial speed with which the gymnast leaves the trampoline.

(which velocity is 0 here?)

Gravitational Potential Energy

Potential energy – stored energy. The higher an object is off of the ground, the more work

gravity can do to it as it falls, and the faster it will be moving as it hits the ground.

Gravitational potential energy is the energy an object has by virtue of its position relative to the surface of the earth.

Measured in Joules.

Potential Energy Example

Calculate and compare the gravitational potential energies of the two objects: Object 1 has a mass of 35 kg and is at a height of 10 m. Object 2 has a mass of 10 kg and is at a height of 35 m.

They have the same potential energy.

Next Class: Quiz on Work and Energy (What we’ve learned so far)

Concepts and problem-solving applications related to: Work done by forces (including gravity and friction) Kinetic and potential energy Work-Energy Theorem

Conservative vs. Nonconservative Forces

Conservative force – total Work on a closed path is zero. (ex: gravity)

Nonconservative force – total Work on a closed path is NOT zero. (ex: friction)

Energy

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-W +W

-W

-W

Gravity- downMotion- up

Friction – leftMotion - right

Friction - rightMotion- left

Gravity- downMotion- down

Conservation of Energy

Law of Conservation of Energy – Energy cannot be created or destroyed, only converted from one form to another.

This means the amount of energy when everything started is still the amount of energy in the universe today! (Just in different forms!)

Conservation of Mechanical Energy

If non-conservative forces (such as friction or air resistance) ARE present:

Be careful: Work done by friction is always negative! (Friction always opposes the motion)

So if friction is present, there is mechanical energy loss. (The energy is converted into heat and sound.)

Conservation of Mechanical Energy

If non-conservative forces are NOT present (or are ignored) the total Mechanical Energy initially is equal to the total Mechanical Energy final.

OR

Conceptual Example 1: Pendulum

Pendulum - Kinetic and Potential Energy In the absence of air resistance and friction…

the pendulum would swing forever example of conservation of mechanical energy Potential → Kinetic → Potential and so on…

In reality, air resistance and friction cause mechanical energy loss, so the pendulum will eventually stop.

Conceptual Example 2: Roller Coaster

Roller Coaster - Kinetic and Potential Energy

Conceptual Example 3: Downhill Skiing

Downhill Skiing - Kinetic and Potential Energy

This animation neglects friction and air resistance until the bottom of the hill.

Friction is provided by the unpacked snow. Mechanical energy loss (nonconservative force) Negative work

Mousetrap Cars

Problem Solving Insights

Determine if non-conservative forces are included. If yes: MEf = ME0 + Wnc

If no: MEf = ME0

Eliminate pieces that are zero before solving Key words: starts from rest (KE0 = 0), ends on the ground

(PEf = 0), etc.

Example 1

The Magnum XL-200 at Cedar Point includes a vertical drop of 59.4m. Assume the roller coaster has a speed of nearly zero at the crest of the hill. Neglecting friction, find the speed of the coaster at the bottom of the hill.

MEf = ME0

KEf + PEf = KE0 + PE0

½ mvf2 + mghf = ½ mv0

2 + mgh0

½ mvf2 = mgh0 (mass cancels!)

vf2 = 2(9.8)(59.4) → vf

= 34.1 m/s

Example 2

A 55.0 kg skateboarder starts out with a speed of 1.80 m/s. He does +80.0J of work on himself by pushing with his feet against the ground. In addition, friction does -265 J of work on him. The final speed of the skateboarder is 6.00 m/s. a) Calculate the change in gravitational potential energy.

b) How much has the vertical height of the skater changed, and is the skater above or below the starting point?

Example 3

A 2.00kg rock is released from rest from a height of 20.0m. Ignore air resistance & determine the kinetic, potential, & mechanical energy at each of the following heights: 20.0m, 12.0m, 0m (Round g to 10 m/s2 for ease)

Energy

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Example 4

Find the potential energy, kinetic energy, mechanical energy, velocity, and height of the skater at the various locations below.

max

Power

Power: Rate of doing work. The work done per unit time.

EquationP = W/t or P =(F d)/t

P is power (Watts, ft lb/s , ft lb/min)

Horsepower: another unit for measuring power.1 horsepower = 746 Watts (or 1 horsepower = 550

ftlb/s) To find horsepower, divide P (in Joules) by 746.

Power Example #1

A weight lifter lifts a 75 kg weight from the ground to a height of 2.0 m. He performs this feat in 1.5 seconds. Find the weight lifter’s average power in A) Watts and B)Horsepower.

Power Example #2

A runner sprints 100 m up a hill in 25 seconds. Her average power during this run is 800 Watts. Find the force that the runner exerts during the run.

Power Example #3

A car accelerates from rest to 20.0 m/s is 5.6 seconds along a level stretch of road. Ignoring friction, determine the average power required to accelerate the car if

A. The weight of the car is 9,000 N

B. The weight of the car is 14,000 N.