copyright © 2012 pearson education inc. energy, work, & power! why does studying physics take...

Copyright © 2012 Pearson Education Inc.

Energy, Work, & Power!

• Why does studying physics take so much energy…. but

• studying physics at your desk is NOT work?


Goals for This Chapter

• Understand & calculate work done by a force

• Understand meaning of kinetic energy

• Learn how work changes kinetic energy of a body & how to use this principle

• Relate work and kinetic energy when forces are not constant or body follows curved path

• To solve problems involving power


Introduction

• The simple methods we’ve learned using Newton’s laws are inadequate when the forces are not constant.

• In this chapter, the introduction of the new concepts of work, energy, and the conservation of energy will allow us to deal with such problems.


Energy, Work, & Power!

• Energy: A new tool in your belt for solving physics problems

• Sometimes, sawing a branch is MUCH easier than hammering it off!

7-1 Kinetic Energy

Energy is required for any sort of motion Energy:

o A scalar quantity assigned to object or system

o Can be changed from one form to another

o Is conserved in a closed system, (total amount of energy of all types is always the same)

Chapter 7: one type of energy (kinetic energy) & one method of transferring energy (work)

© 2014 John Wiley & Sons, Inc. All rights reserved.

7-1 Kinetic Energy

o Faster an object moves, greater its kinetic energy

o Kinetic energy is zero for a stationary object

For an object with v well below the speed of light:

The unit of kinetic energy is a joule (J)


7-1 Kinetic Energy

Example Energy of 2200 lb car moving 65 mph?

Mass = 1000 kg

Velocity = ~ 30 m/s

KE = 4.4 x 105 J


7-1 Kinetic Energy

Example Energy of 18-wheel semi moving 65 mph?

Mass = ~ 67000 kg

Velocity = ~ 30 m/s

KE = 6.0 x 107 J


7-1 Kinetic Energy

Example Energy dissipated in collision?

4.4 x 105 J + 6.0 x 107 J = ~ 6.1 x 107 J


7-1 Kinetic Energy

Example

• Force of Truck on Car?

•Force of Car on Truck?

•ACCELERATION of car? Of Truck?


7-2 Work and Kinetic Energy

Account for changes in kinetic energy by saying energy has been transferred to or from the object

In a transfer of energy via an external force, work is:o Done on the object by the force



This is not the common meaning of the word “work”o To do work on an object, energy must be transferredo Throwing a baseball does worko Pushing an immovable wall does not do work



Start from force equation and 1-dimensional velocity:

Rearrange into kinetic energies:

Left side is change in energy Work is:



Work

• A force on a body does work only if the body undergoes a displacement.


Work done by a constant force

• The work done by a constant force acting at an angle to the displacement is

W = Fs cos .


Work done by a constant force

• The work done by a constant force acting at an angle to the displacement is

W = Fs cos .

• Units of Work = Force x Distance = Newtons x meters

= (N m) = (kg m/s2) m = JOULE of energy!

• Work has the SI unit of joules (J), the same as energy

• In the British system, the unit is foot-pound (ft lb)


Positive, negative, and zero work

• A force can do positive, negative, or zero work depending on the angle between the force and the displacement.


For an angle φ between force and displacement:

As vectors we can write:

Notes on these equations:o Force is constanto Object is particle-like (rigid)o Work can be positive or negative





• Work is done BY an external force, ON an object.

•Positive work done by an external force





•Positive work done by an external force (with no other forces acting in that direction of motion)






will speed up an object!






will speed up an object!

•Negative work done by an external force (wnofaitdom) will slow down an object!


For two or more forces, net work is sum of work done by all individual external forces

Two methods to calculate net work: Find all work and sum individual terms

OR….

Take vector sum of forces (Fnet

) and calculate net work

…..once



Work done by several forces – Example

• Tractor pulls wood 20 m over level ground;

• Weight = 14,700 N; Tractor exerts 5000 N force at 36.9 degrees above horizontal.

• 3500 N friction opposes!



• How much work is done BY the tractor ON the sled with wood?

• How much work is done BY gravity ON the sled?

• How much work is done BY friction ON the sled?



• Tractor pulls wood 20 m over level ground; w = 14,700 N; Tractor exerts 5000 N force at 36.9 degrees above horizontal. 3500 N friction opposes!


Kinetic energy

• The kinetic energy of a particle is KE = 1/2 mv2.

• Net work on body changes its speed & kinetic energy


Kinetic energy

• The kinetic energy of a particle is K = 1/2 mv2.

• Net work on body changes its speed & kinetic energy


The work-kinetic energy theorem states:

(change in kinetic energy) = (net work done)

Or we can write it as:

(final KE) = (initial KE) + (net work)



The work-energy theorem

• The work done by the net force on a particle equals the change in the particle’s kinetic energy.

• Mathematically, the work-energy theorem is expressed as

Wtotal = KEfinal – KEinitial = KE



• Tractor pulls wood 20 m over level ground; w = 14,700 N; Tractor exerts 5000 N force at 36.9 degrees above horizontal. 3500 N friction opposes!

• Suppose sled moves at 20 m/s; what is speed after 20 m?


Using work and energy to calculate speed


The work-kinetic energy theorem holds for positive and negative work

Example If the kinetic energy of a particle is initially 5 J:

o A net transfer of 2 J to the particle (positive work)

• Final KE = 7 J

o A net transfer of 2 J from the particle (negative work)

• Final KE = 3 J



Answer: (a) energy decreases (b) energy remains the same

(c) work is negative for (a), and work is zero for (b)



Comparing kinetic energies – example

• Two iceboats have different masses, m & 2m. Wind exerts same force; both start from rest.


Comparing kinetic energies – example 6.5

• Two iceboats have different masses, m & 2m. Wind exerts same force; both start from rest. Which boat wins? Which boat crosses with the most KE?

7-3 Work Done by the Gravitational Force

We calculate the work as we would for any force Our equation is:

For a rising object:

For a falling object:

Eq. (7-12)

Eq. (7-14)

Eq. (7-13)



Forces on a hammerhead – example

• Hammerhead of pile driver drives a beam into the ground.

Find speed as it hits and average force on the beam.


Forces on a hammerhead – example 6.4

• Hammerhead of pile driver drives a beam into the ground, following guide rails that produce 60 N of frictional force


Orientations of forces and associated works for upward displacement

Note works need not be equal, they are only equal if initial and final kinetic energies are equal

If works are unequal, you need to know difference between initial & final kinetic energy to solve for work



Orientations of forces and their associated works for downward displacement



Figure 7-8

Examples You are a passenger:o Being pulled up a ski-slope

• Tension does positive work, gravity does negative work



Examples You are a passenger:

o Being lowered down in an elevator

• Tension does negative work,

• Gravity does positive work



Work and energy with varying forces

• Many forces, such as force to stretch a spring, are not constant.

7-4 Work Done by a Spring Force

Spring force is a variable force from a springo Particular mathematical formo Many forces in nature have this form

Figure (a) shows spring in a relaxed state: neither compressed nor extended, no force is applied

Stretch or extend spring? It resists & exerts a restoring force that attempts to return the spring to its relaxed state



Spring force is given by Hooke's law:

Negative sign means force always opposes displacement

Spring constant k measures stiffness of the spring

This is a variable force (function of position) and it exhibits a linear relationship between F and d

For a spring along the x-axis we can write:



Stretching a spring

• The force required to stretch a spring a distance x is proportional to x:

Fx = kx

• k is the force constant(or spring constant)

• Units of k = Newtons/meter

– Large k = TIGHT spring

– Small k = L O O S E spring


Work and energy with varying forces

• Suppose an applied force (like that of a stretched spring) is not constant.


Work and energy with varying forces—Figure 6.16

• Approximate work by dividing total displacement into many small segments.



• Work = ∫ F∙dx

• An infinite summation of tiny rectangles!



• Work = ∫ F∙dx

• An infinite summation of tiny rectangles!

• Height: F(x)Width: dxArea: F(x)dx

• Total area: ∫ F∙dx


We can find the work by integrating:

Plug kx in for Fx:

Work:o Can be positive or negativeo Depends on the net energy transfer



The work:o Can be positive or negative



For an initial position of x = 0:


Area under graph represents work done on the spring to stretch it a distance X:

W = ½ kX2


For an applied force where the initial and final kinetic energies are zero:



Answer: (a) positive

(b) negative

(c) zero


7-5 Work Done by a General Variable Force

We take a one-dimensional example

We need to integrate the work equation (which normally applies only for a constant force) over the change in position

We can show this process by an approximation with rectangles under the curve

Figure 7-12


7-5 Work Done by a General Variable Force

Our sum of rectangles would be:

As an integral this is:

In three dimensions, we integrate each separately:

The work-kinetic energy theorem still applies!

Eq. (7-31)

Eq. (7-32)

Eq. (7-36)


7-6 Power

7.18 Apply the relationship between average power, the work done by a force, and the time interval in which that work is done.

7.19 Given the work as a function of time, find the instantaneous power.

7.20 Determine the instantaneous power by taking a dot product of the force vector and an object's velocity vector, in magnitude-angle and unit-vector notations.

Learning Objectives


7-6 Power

Power is the time rate at which a force does work A force does W work in a time Δt; the average power

due to the force is:

The instantaneous power at a particular time is:

The SI unit for power is the watt (W): 1 W = 1 J/s Therefore work-energy can be written as (power) x

(time) e.g. kWh, the kilowatt-hour

Eq. (7-42)

Eq. (7-43)


7-6 Power

Solve for the instantaneous power using the definition of work:

Or:Eq. (7-47)

Eq. (7-48)

Answer: zero (consider P = Fv cos ɸ, and note that ɸ = 90°)


Kinetic Energy The energy associated with

motion

Work Energy transferred to or from

an object via a force

Can be positive or negative

7 Summary

Eq. (7-1)

Eq. (7-7)Eq. (7-10)

Work Done by a Constant Force

The net work is the sum of individual works

Work and Kinetic Energy

Eq. (7-11)Eq. (7-8)


Work Done by the Gravitational Force

Work Done in Lifting and Lowering an Object

Eq. (7-16)

Eq. (7-26)

7 Summary

Eq. (7-12)

Spring Force Relaxed state: applies no

force

Spring constant k measures stiffness

Eq. (7-20)

Spring Force For an initial position x = 0:


Work Done by a Variable Force

Found by integrating the constant-force work equation

Power The rate at which a force

does work on an object

Average power:

Instantaneous power:

For a force acting on a moving object:

7 Summary

Eq. (7-32)

Eq. (7-43)

Eq. (7-47)

Eq. (7-42)

Eq. (7-48)



Work done on a spring scale – example 6.6

• A woman of 600 N weight compresses spring 1.0 cm. What is k and total work done BY her ON the spring?


Work done on a spring scale – example 6.6

• A woman of 600 N weight compresses spring 1.0 cm. What is k and total work done?


Motion with a varying force

• An air-track glider mass 0.1 kg is attached to a spring of force constant 20 N/m, so the force on the glider is varying.

• Moving at 1.50 m/s to right when spring is unstretched.

• Find maximum distance moved if no friction, and if friction was present with k = 0.47



• glider mass 0.1 kg

• spring of force constant 20 N/m

• Moving at 1.50 m/s to right when spring is unstretched.

• Final velocity once spring stops glider = 0

• We know 3 things! F/m = a; vi; vf

• Why can’t we use F = ma ?



• An air-track glider is attached to a spring, so the force on the glider is varying.

• In general, if the force varies, using ENERGY will be an easier method than using forces!

• We know:

• Initial KE = ½ mv2

• Final KE = ½ mvf2 = 0

• Get net work done!

• Get ½ kx2



• An air-track glider is attached to a spring, so the force on the glider is varying.


Motion on a curved path—Example 6.8

• A child on a swing moves along a smooth curved path at constant speed.

• Weight w, Chain length = R, max angle = 0

• You push with force F that varies.

• What is work done by you?

• What is work done by gravity?

• What is work done by chain?



• A child on a swing moves along a curved path.

• Weight w, Chain length = R, max angle = 0

• You push with force F that varies.



• W = ∫ F∙dl

• F∙dl = F cos

• |dl| = ds (distance along the arc)


Power

• Power is rate work is done.

• Average power is Pav = W/t

• Instantaneous power is P = dW/dt.

• SI unit of power is watt (1 W = 1 J/s)

• horsepower (1 hp = 746 W ~ ¾ of kilowatt)

• kilowatt-hour (kwh) is ENERGY (power x time)


Force and power

• Jet engines develop power to fly the plane.


A “power climb”

• A person runs up stairs

copyright © 2012 pearson education inc. energy, work, & power! why does studying physics take...

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