Physics 215 – Fall 2014 Lecture 09-1 1

Welcome back to Physics 215

Today’s agenda:

• Work

• Non-conservative forces

• Power

Physics 215 – Fall 2014 Lecture 09-1 2

Current homework assignment

• HW7:– Knight Textbook Ch.9: 54, 72– Ch.10: 48, 68, 76– Ch.11: 50, 64– Due Wednesday, Oct. 22nd in recitation

Physics 215 – Fall 2014 Lecture 09-1 3

Work, Energy

• Newton’s Laws are vector equations

• Sometimes easier to relate speed of a particle to how far it moves under a force – a single equation can be used – need to introduce concept of work

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What is work?

• Assume constant force in 1D

• Consider: vF

2 = vI2 + 2a s

• Multiply by m/2 (1/2)mvF

2 - (1/2)mvI2 = m a s

• But: F = ma

(1/2)mvF2 - (1/2)mvI

2 = F s

Physics 215 – Fall 2014 Lecture 09-1 5

Work-Kinetic Energy theorem (1)

(1/2)mvF2 - (1/2)mvI

2 = F s

Points:

• W = Fs defines work done on particle

= force times displacement

• K = (1/2)mv2 defines kinetic energy

=1/2 mass times square of v

Physics 215 – Fall 2014 Lecture 09-1 6

Work-Kinetic Energy demo

• Cart, force probe, and motion detector

• Plot v2 vs. x – gradient 2F/m

• constant F (measure) -- pulling with string

• Weigh cart and masses in advance

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Conclusions from experiment

Although the motion of the two carts looks very different (i.e., different amounts of time, accelerations, and final speeds), there is a quantity that is the same for both at the end of the motion. It is (1/2) mv2 and is called the (final) kinetic energy of the carts.

Moreover, this quantity happens to have the same value as F s, which is given the name work.

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Improved definition of work

• For forces, write F FAB

• Thus W = F s WAB = FAB sA is work done on A by B as A undergoes

displacement sA

• Work-kinetic energy theorem:

Wnet,A = BWAB = K

Physics 215 – Fall 2014 Lecture 09-1 9

Non-constant force …

x

F

x

F(x)

Work done in small interval xW = F x

Total W done from A to B

F x =Area under curve!

BA

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The net work done on an object is equal to the change in kinetic energy of the object.

The Work - Kinetic Energy Theorem

Wnet = K = Kf - Ki

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1. It takes equal distances to stop each ball.

2. It takes equal time intervals to stop each ball.

3. Both of the above.

4. Neither of the above.

Suppose a tennis ball and a bowling ball are rolling toward you. The tennis ball is moving much faster, but both have the same momentum (mv), and you exert the same force to stop each.

Which of the following statements is correct?

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1. less than.

2. equal to

3. greater than

Suppose a tennis ball and a bowling ball are rolling toward you. Both have the same momentum (mv), and you exert the same force to stop each.

The distance taken for the bowling ball to stop is

the distance taken for the tennis ball to stop.

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Two carts of different mass are accelerated from rest on a low-friction track by the same force for the same time interval.

Cart B has greater mass than cart A (mB > mA). The final speed of cart A is greater than that of cart B (vA > vB).

After the force has stopped acting on the carts, the kinetic energy of cart B is

1. less than the kinetic energy of cart A (KB < KA).

2. equal to the kinetic energy of cart A (KB = KA).

3. greater than the kinetic energy of cart A (KB > KA).

4. “Can’t tell.”

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Kinetic energy of an object:

Work done on object 1 by object 2:

Revised definitions for Work and Kinetic Energy

Physics 215 – Fall 2014 Lecture 09-1 15

Scalar (or “dot”) product of vectors

The scalar product is a way to combine two vectors to obtain a number (or scalar). It is indicated by a dot (•) between the two vectors.

(Note: component of A in direction n is just A•n)

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1. positive

2. negative

3. equal to zero

4. “Can’t tell.”

Is the scalar (“dot”) product of the two vectors

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1. positive

2. negative

3. equal to zero

4. “Can’t tell.”

Is the scalar (“dot”) product of the two vectors

Physics 215 – Fall 2014 Lecture 09-1 18

Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B.

The speed of block A at the bottom of its ramp is

1. less than the speed of block B.

2. equal to the speed of block B.

3. greater than the speed of block B.

4. “Can’t tell.”

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Solution

• Which forces do work on block?

• Which, if any, are constant?

• What is F•s for motion?

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Work done by gravity

N

smg

Work W = -mg j•s

Therefore,W = -mgh

N does no work!

i

j

Physics 215 – Fall 2014 Lecture 09-1 21

Work done on an object by gravity

W (on object by earth) = – m g h,

where h = hfinal – hinitial is the change in height.

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Defining gravitational potential energy

The change in gravitational potential energy of the object-earth system is just another name for the negative value of the work done on an object by the earth.

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Work done by gravity between 2fixed pts does not depend on pathtaken!

Curved ramps =

W = F•s =

Work done by gravitational force in moving someobject along any path is independent of the pathdepending only on the change in vertical height

Physics 215 – Fall 2014 Lecture 09-1 24

Hot wheels demoOne hot wheels car, car A, rolls down the incline and travels straight ahead, while the other car, B, goes through a loop at the bottom of the incline. When the cars reach the end of their respective tracks, the relative speeds will be:

1. vA > vB

2. vA < vB

3. vA = vB

4. Can’t tell

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1. positive

2. negative

3. equal to zero

4. “Can’t tell.”

A person carries a book horizontally at constant speed. The work done on the book by the person’s hand is

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1. negative and equal in absolute value to W1

2. negative and less in absolute value than W1

3. positive and equal in absolute value to W1

4. positive and less in absolute value than W1

A person lifts a book at constant speed. Since the force exerted on the book by the person’s hand is in the same direction as the displacement of the book, the work (W1) done on the book by the person’s hand is positive.

The work done on the book by the earth is:

Physics 215 – Fall 2014 Lecture 09-1 27

A person lifts a book at constant speed. The work (W1) done on the book by the person’s hand is positive.

Work done on the book by the earth:

Net work done on the book:

Change in kinetic energy of the book:

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Work-Kinetic Energy theorem(final statement)

W1net =Fnet •s1 = Kf - Ki

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1. Yes, positive work.

2. Yes, negative work.

3. No, zero work.

4. No, since the lower book does work on the upper book, this is not a meaningful question.

A person lifts two books (each of mass m) at constant speed. The work done on the upper book by the lower book is positive. Its magnitude is W = m g s.

Is there work done on the lower book by the upper book?

Physics 215 – Fall 2014 Lecture 09-1 30

A person lifts two books at constant speed. The work done on the upper book by the lower book is positive.

Work done on lower book by upper book:

Net work done on the lower book:

Change in kinetic energy of the lower book:

Physics 215 – Fall 2014 Lecture 09-1 31

Conservative forces

• If the work done by some force (e.g. gravity) does not depend on path the force is called conservative.

• Then gravitational potential energy Ug only depends on (vertical) position of object Ug = Ug(h)

• Elastic forces also conservative – elastic potential energy U = (1/2)kx2...

Physics 215 – Fall 2014 Lecture 09-1 32

Nonconservative forces

• friction, air resistance,...

• Potential energies can only be defined for conservative forces

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• Can do work, but cannot be represented by a potential energy function

• Total mechanical energy can now change due to work done by nonconservative force

• For example, frictional force leads to decrease of total mechanical energy -- energy converted to heat, or internal energy

• Total energy = total mech. energy + internal energy -- is conserved

Nonconservative forces

Physics 215 – Fall 2014 Lecture 09-1 34

Conservation of (mechanical) energy

• If we are dealing with a potential energy corresponding to a conservative force

0 = K+U

• Or K + U = constant

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Conservation of total energy

The total energy of an object or system is said to be conserved if the sum of all energies (including those outside of mechanics that have not yet been discussed) never changes.

This is believed always to be true.

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Demo

• Damping from air resistance

• Amplitude of oscillations decays

• Oscillations still about x = -xeq

Vertical spring and mass with damping

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Many forces

• For a particle which is subject to several (conservative) forces F1, F2 …

E = (1/2)mv2 + U1 + U2 +… is constant

• Principle called

Conservation of total mechanical energy

Physics 215 – Fall 2014 Lecture 09-1 38

Summary

• Total (mechanical) energy of an isolated system is constant in time.

• Must be no non-conservative forces

• Must sum over all conservative forces

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A compressed spring fires a ping pong ball vertically upward. If the spring is compressed by 1 cm initially the ball reaches a height of 2 m above the spring. What height would the ball reach if the spring were compressed by just 0.5 cm? (neglect air resistance)

1. 2 m

2. 1 m

3. 0.5 m

4. we do not have sufficient information to calculate the new height

Physics 215 – Fall 2014 Lecture 09-1 40

A 5.00-kg package slides 1.50 m down a long ramp that is inclined at 12.0 below the horizontal. The coefficient of kinetic friction between the package and the ramp is k = 0.310. Calculate: (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after sliding 1.50 m down the ramp?

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Summary

• Work is defined as dot product of force with displacement vector

• Each individual force on an object gives rise to work done

• The kinetic energy only changes if net work is done on the object, which requires a net force

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Power

Power = Rate at which work is done

Units of power:

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1. 2.25 s

2. 3.0 s

3. 4.5 s

4. 6.0 s

A sports car accelerates from zero to 30 mph in 1.5 s. How long does it take to accelerate from zero to 60 mph, assuming the power (=W / t) of the engine to be constant?

(Neglect losses due to friction and air drag.)

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Power in terms of force and velocity

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1. decreasing

2. constant

3. increasing

A locomotive accelerates a train from rest to a final speed of 40 mph by delivering constant power. If we assume that there are no losses due to air drag or friction, the acceleration of the train (while it is speeding up) is

Physics 215 – Fall 2014 Lecture 09-1 46

1. positive

2. negative

3. zero

4. “Can’t tell.”

A ball is whirled around a horizontal circle at constant speed.

If air drag forces can be neglected, the power expended by the hand is:

Physics 215 – Fall 2014 Lecture 09-1 47

Reading assignment

• Extended objects, center of mass

• Begin Chapter 12 in textbook