Physics 215 – Fall 2014 Lecture 08-2 1

Welcome back to Physics 215

Today’s agenda:

• More gravitational potential energy

• Potential energy of a spring

• Work-kinetic energy theorem

Physics 215 – Fall 2014 Lecture 08-2 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 08-2 3

Gravitational Potential Energy

For an object of mass m near the surface of the earth:

Ug = mgh

• h is height above arbitrary reference line• Measured in Joules -- J (like kinetic energy)

Physics 215 – Fall 2014 Lecture 08-2 4

Total energy for object moving under gravity

E = Ug + K = constant

* E is called the (mechanical) energy

* It is conserved:

(½) mv2 + mgh = constant

Physics 215 – Fall 2014 Lecture 08-2 5

A ball of mass m=7 kg attached to a massless string of length R=3 m is released from the position shown in the figure below. (a) Find magnitude of velocity of the ball at the lowest point on its path. (b) Find the tension in the string at that point.

Physics 215 – Fall 2014 Lecture 08-2 6

Stopped-pendulum demo

• Pendulum swings to same height on other side of vertical

• What if pendulum string is impeded ~1/2-way along its length? Will height on other side of vertical be:

1. Greater than original height2. Same as original height3. Less than original height?

Physics 215 – Fall 2014 Lecture 08-2 7

A block is released from rest on a frictionless incline. The block travels to the bottom of the left incline and then moves up the right incline which is steeper than the left side.

Physics 215 – Fall 2014 Lecture 08-2 8

Springs -- Elastic potential energy

Force F = -kx (Hooke’s law)

Area of triangle lying under straight line graph of F vs. x = (1/2)(+/-x)(-/+kx)

F

xF = -kxUs = (1/2)kx2

frictionless table

Physics 215 – Fall 2014 Lecture 08-2 9

(Horizontal) Spring

frictionless table

(1/2)kx2 + (1/2)mv2 = constant

• x = displacement from relaxed state of spring

• Elastic potential energy stored in spring: Us = (1/2)kx2

Physics 215 – Fall 2014 Lecture 08-2 10

A 0.5 kg mass is attached to a spring on a horizontal frictionless table. The mass is pulled to stretch the spring 5.0 cm and is released from rest. When the mass crosses the point at which the spring is not stretched, x = 0, its speed is 20 cm/s. If the experiment is repeated with a 10.0 cm initial stretch, what speed will the mass have when it crosses x = 0 ?

1. 40 cm/s

2. 0 cm/s

3. 20 cm/s

4. 10 cm/s

Physics 215 – Fall 2014 Lecture 08-2 11

Mass hanging on spring

• Now oscillations are about equilibrium point of spring + mass

• Otherwise, motion is same as horizontal mass + spring on frictionless table

(1/2)mv2 = (1/2)ka2 - (1/2)kz2

Physics 215 – Fall 2014 Lecture 08-2 12

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

Physics 215 – Fall 2014 Lecture 08-2 13

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 08-2 14

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 08-2 15

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

Physics 215 – Fall 2014 Lecture 08-2 16

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.

Physics 215 – Fall 2014 Lecture 08-2 17

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 08-2 18

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

Physics 215 – Fall 2014 Lecture 08-2 19

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

Physics 215 – Fall 2014 Lecture 08-2 20

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?

Physics 215 – Fall 2014 Lecture 08-2 21

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.

It takes equal time intervals to stop each ball.

The distance taken for the bowling ball to stop is

the distance taken for the tennis ball to stop.

Physics 215 – Fall 2014 Lecture 08-2 22

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.”

Physics 215 – Fall 2014 Lecture 08-2 23

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 08-2 24

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

Physics 215 – Fall 2014 Lecture 08-2 26

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 08-2 27

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.”

Physics 215 – Fall 2014 Lecture 08-2 28

Solution

• Which forces do work on block?

• Which, if any, are constant?

• What is F•s for motion?

Physics 215 – Fall 2014 Lecture 08-2 29

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 08-2 30

Reading assignment

• More W-KE Theorem

• Conservative and non-conservative forces

• Power

• Finish chapter 11 in textbook