Energy, Work, & Power! Why does studying physics take so much energy…. but studying physics at your desk is NOT work? 

Understand & calculate work done by a force 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!

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-1 Kinetic Energy Energy is required for any sort of motion Energy: A scalar quantity assigned to object or system Can be changed from one form to another 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. 5

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-1 Kinetic Energy Faster an object moves, greater its kinetic energy 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) © 2014 John Wiley & Sons, Inc. All rights reserved. 6

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

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

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-1 Kinetic Energy Example Energy dissipated in collision? 4.4 x 105 J + 6.0 x 107 J = ~ 6.1 x 107 J © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-1 Kinetic Energy Example Force of Truck on Car? Force of Car on Truck? ACCELERATION of car? Of Truck? © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. 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: Done on the object by the force © 2014 John Wiley & Sons, Inc. All rights reserved. 11

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy This is not the common meaning of the word “work” To do work on an object, energy must be transferred Throwing a baseball does work Pushing an immovable wall does not do work © 2014 John Wiley & Sons, Inc. All rights reserved. 12

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy Start from force equation and 1-dimensional velocity: Rearrange into kinetic energies: Left side is change in energy Work is: © 2014 John Wiley & Sons, Inc. All rights reserved. 13

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.

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

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

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy For an angle φ between force and displacement: As vectors we can write: Notes on these equations: Force is constant Object is particle-like (rigid) Work can be positive or negative © 2014 John Wiley & Sons, Inc. All rights reserved. 20

Positive, negative, and zero work A force can do positive, negative, or zero work depending on the angle between the force and the displacement. Work is done BY an external force, ON an object. Positive work done by an external force

Positive, negative, and zero work A force can do positive, negative, or zero work depending on the angle between the force and the displacement. Work is done BY an external force, ON an object. Positive work done by an external force (with no other forces acting in that direction of motion)

Positive, negative, and zero work A force can do positive, negative, or zero work depending on the angle between the force and the displacement. Work is done BY an external force, ON an object. Positive work done by an external force (with no other forces acting in that direction of motion) will speed up an object!

Positive, negative, and zero work A force can do positive, negative, or zero work depending on the angle between the force and the displacement. Work is done BY an external force, ON an object. Positive work done by an external force (with no other forces acting in that direction of motion) will speed up an object! Negative work done by an external force (wnofaitdom) will slow down an object!

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy 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  © 2014 John Wiley & Sons, Inc. All rights reserved. 26

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!

Work done by several forces – Example 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?

Work done by several forces – Example 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

Kinetic energy The kinetic energy of a particle is K = 1/2 mv2. Net work on body changes its speed & kinetic energy

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and 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) © 2014 John Wiley & Sons, Inc. All rights reserved. 33

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

Work done by several forces – Example 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

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy The work-kinetic energy theorem holds for positive and negative work Example If the kinetic energy of a particle is initially 5 J: A net transfer of 2 J to the particle (positive work) Final KE = 7 J A net transfer of 2 J from the particle (negative work) Final KE = 3 J © 2014 John Wiley & Sons, Inc. All rights reserved. 37

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy © 2014 John Wiley & Sons, Inc. All rights reserved. 38

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-2 Work and Kinetic Energy Answer: (a) energy decreases (b) energy remains the same (c) work is negative for (a), and work is zero for (b) © 2014 John Wiley & Sons, Inc. All rights reserved. 39

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?

© 2014 John Wiley & Sons, Inc. All rights reserved. 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. (7-13) Eq. (7-14) © 2014 John Wiley & Sons, Inc. All rights reserved. 42

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

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

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-3 Work Done by the Gravitational 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 © 2014 John Wiley & Sons, Inc. All rights reserved. 46

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-3 Work Done by the Gravitational Force Orientations of forces and their associated works for downward displacement © 2014 John Wiley & Sons, Inc. All rights reserved. 47

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-3 Work Done by the Gravitational Force Examples You are a passenger: Being pulled up a ski-slope Tension does positive work, gravity does negative work Figure 7-8 © 2014 John Wiley & Sons, Inc. All rights reserved. 48

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-3 Work Done by the Gravitational Force Examples You are a passenger: Being lowered down in an elevator Tension does negative work, Gravity does positive work © 2014 John Wiley & Sons, Inc. All rights reserved. 49

Work and energy with varying forces Many forces, such as force to stretch a spring, are not constant.

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force Spring force is a variable force from a spring Particular mathematical form 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 © 2014 John Wiley & Sons, Inc. All rights reserved. 51

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force 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: © 2014 John Wiley & Sons, Inc. All rights reserved. 52

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 and energy with varying forces—Figure 6.16 Work = ∫ F∙dx An infinite summation of tiny rectangles!

Work and energy with varying forces—Figure 6.16 Work = ∫ F∙dx An infinite summation of tiny rectangles! Height: F(x) Width: dx Area: F(x)dx Total area: ∫ F∙dx

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force We can find the work by integrating: Plug kx in for Fx: Work: Can be positive or negative Depends on the net energy transfer © 2014 John Wiley & Sons, Inc. All rights reserved. 58

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force The work: Can be positive or negative © 2014 John Wiley & Sons, Inc. All rights reserved. 59

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force 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 © 2014 John Wiley & Sons, Inc. All rights reserved. 60

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force For an applied force where the initial and final kinetic energies are zero: © 2014 John Wiley & Sons, Inc. All rights reserved. 61

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force © 2014 John Wiley & Sons, Inc. All rights reserved. 62

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-4 Work Done by a Spring Force Answer: (a) positive (b) negative (c) zero © 2014 John Wiley & Sons, Inc. All rights reserved. 63

© 2014 John Wiley & Sons, Inc. All rights reserved. 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 © 2014 John Wiley & Sons, Inc. All rights reserved. 64

© 2014 John Wiley & Sons, Inc. All rights reserved. 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) © 2014 John Wiley & Sons, Inc. All rights reserved. 65

© 2014 John Wiley & Sons, Inc. All rights reserved. 7-6 Power Learning Objectives 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. © 2014 John Wiley & Sons, Inc. All rights reserved. 66

© 2014 John Wiley & Sons, Inc. All rights reserved. 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) © 2014 John Wiley & Sons, Inc. All rights reserved. 67

© 2014 John Wiley & Sons, Inc. All rights reserved. 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°) © 2014 John Wiley & Sons, Inc. All rights reserved. 68

© 2014 John Wiley & Sons, Inc. All rights reserved. 7 Summary Kinetic Energy The energy associated with motion Work Energy transferred to or from an object via a force Can be positive or negative Eq. (7-1) Work Done by a Constant Force The net work is the sum of individual works Work and Kinetic Energy Eq. (7-10) Eq. (7-7) Eq. (7-8) Eq. (7-11) © 2014 John Wiley & Sons, Inc. All rights reserved. 69

© 2014 John Wiley & Sons, Inc. All rights reserved. 7 Summary Work Done by the Gravitational Force Work Done in Lifting and Lowering an Object Eq. (7-12) Eq. (7-16) Spring Force Relaxed state: applies no force Spring constant k measures stiffness Spring Force For an initial position x = 0: Eq. (7-26) Eq. (7-20) © 2014 John Wiley & Sons, Inc. All rights reserved. 70

© 2014 John Wiley & Sons, Inc. All rights reserved. 7 Summary 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: Eq. (7-42) Eq. (7-32) Eq. (7-43) Eq. (7-47) Eq. (7-48) © 2014 John Wiley & Sons, Inc. All rights reserved. 71

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 mk = 0.47

Motion with a varying force 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 ?

Motion with a varying force 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

Motion with a varying force 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 = q0 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?

Motion on a curved path—Example 6.8 A child on a swing moves along a curved path. Weight w, Chain length = R, max angle = q0 You push with force F that varies.

Motion on a curved path—Example 6.8 W = ∫ F∙dl F∙dl = F cos q |dl| = ds (distance along the arc)

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

Jet engines develop power to fly the plane. Force and power Jet engines develop power to fly the plane.

A “power climb” A person runs up stairs