Loading...

PPT – Goals: PowerPoint presentation | free to download - id: 704bbd-MDA2N

The Adobe Flash plugin is needed to view this content

Lecture 15

- Goals

- Chapter 11
- Employ conservative and non-conservative forces
- Use the concept of power (i.e., energy per time)
- Chapter 12
- Extend the particle model to rigid-bodies
- Understand the equilibrium of an extended

object. - Understand rotation about a fixed axis.
- Employ conservation of angular momentum concept

- Assignment
- HW7 due March 25th
- For Thursday Read Chapter 12, Sections 7-11
- do not concern yourself with the integration

process in regards to center of mass or moment

of inertia

More Work 2-D Example (constant force)

- An angled force, F 10 N, pushes a box across a

frictionless floor for a distance ?x 5 m and ?y

0 m

Finish

Start

F

q -45

Fx

?x

- (Net) Work is Fx ?x F cos(-45) ?x 50 x

0.71 Nm 35 J - Notice that work reflects energy transfer

Net Work 1-D 2nd Example (constant force)

- A force F 10 N is opposite the motion of a box

across a frictionless floor for a distance ?x 5

m.

Finish

Start

q 180

F

?x

- Net Work is F ?x -10 x 5 N m -50 J
- Work reflects energy transfer

Work in 3D.(assigning U to be external to the

system)

- x, y and z with constant F

A tool Scalar Product (or Dot Product)

A B A B cos(q)

- Useful for performing projections.

A ? î Ax î ? î 1 î ? j 0

You choose the way that works best for you!

Scalar Product (or Dot Product)

- Compare
- A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
- Redefine A ? F (force), B ? Dr (displacement)
- Notice
- F ? Dr (Fx )(Dx) (Fy )(Dz ) (Fz )(Dz)
- So here
- F ? Dr W
- More generally a Force acting over a Distance

does Work

Definition of Work, The basics

Ingredients Force ( F ), displacement ( ? r )

Work, W, of a constant force F acts through a

displacement ? r W F ? r (Work is a scalar)

F

? r

?

displacement

If we know the angle the force makes with the

path, the dot product gives us F cos q and

Dr If the path is curved at each point and

Remember that a real trajectory implies forces

acting on an object

path and time

Fradial

Ftang

F

0

Two possible options

0

Change in the magnitude of

Change in the direction of

0

- Only tangential forces yield work!
- The distance over which FTang is applied Work

Definition of Work, The basics

Ingredients Force ( F ), displacement ( ? r )

Work, W, of a constant force F acts through a

displacement ? r W F ? r (Work is a scalar)

Work tells you something about what happened on

the path! Did something do work on you? Did

you do work on something? If only one force

acting Did your speed change?

Exercise Work in the presence of friction and

non-contact forces

- A box is pulled up a rough (m gt 0) incline by a

rope-pulley-weight arrangement as shown below. - How many forces (including non-contact ones) are

doing work on the box ? - Of these which are positive and which are

negative? - State the system (here, just the box)
- Use a Free Body Diagram
- Compare force and path

- 2
- 3
- 4
- 5

Exercise Work in the presence of friction and

non-contact forces

- A box is pulled up a rough (m gt 0) incline by a

rope-pulley-weight arrangement as shown below. - How many forces are doing work on the box ?
- And which are positive (T) and which are

negative (f, mg)? - (For mg only the component along the surface is

relevant) - Use a Free Body Diagram

- (A) 2
- (B) 3 is correct
- 4
- 5

v

N

T

f

mg

Work and Varying Forces (1D)

Area Fx Dx F is increasing Here W F ? r

becomes dW F dx

- Consider a varying force F(x)

Fx

x

Dx

Finish

Start

F

F

q 0

Dx

Work has units of energy and is a scalar!

Example Hookes Law Spring (xi equilibrium)

- How much will the spring compress (i.e. ?x xf -

xi) to bring the box to a stop (i.e., v 0 ) if

the object is moving initially at a constant

velocity (vi) on frictionless surface as shown

below and xi is the equilibrium position of the

spring?

Example Hookes Law Spring

- How much will the spring compress to bring the

box to a stop (i.e., v 0 ) if the object is

moving initially at a constant velocity (vi) on

frictionless surface as shown below and xe is the

equilibrium position of the spring? (More

difficult)

Work signs

Notice that the spring force is opposite the

displacement For the mass m, work is

negative For the spring, work is positive

They are opposite, and equal (spring is

conservative)

Conservative Forces Potential Energy

- For any conservative force F we can define a

potential energy function U in the following way - The work done by a conservative force is equal

and opposite to the change in the potential

energy function. - This can be written as

ò

W F dr - ?U

Conservative Forces and Potential Energy

- So we can also describe work and changes in

potential energy (for conservative forces) - DU - W
- Recalling (if 1D)
- W Fx Dx
- Combining these two,
- DU - Fx Dx
- Letting small quantities go to infinitesimals,
- dU - Fx dx
- Or,
- Fx -dU / dx

Exercise Work Done by Gravity

- An frictionless track is at an angle of 30 with

respect to the horizontal. A cart (mass 1 kg) is

released from rest. It slides 1 meter downwards

along the track bounces and then slides upwards

to its original position. - How much total work is done by gravity on the

cart when it reaches its original position? (g

10 m/s2)

1 meter

30

(A) 5 J (B) 10 J (C) 20 J (D) 0 J

Home Exercise Work Friction

- Two blocks having mass m1 and m2 where m1 gt m2.

They are sliding on a frictionless floor and have

the same kinetic energy when they encounter a

long rough stretch (i.e. m gt 0) which slows them

down to a stop. - Which one will go farther before stopping?
- Hint How much work does friction do on each

block ?

(A) m1 (B) m2 (C) They will go the same

distance

m1

v1

v2

m2

Exercise Work Friction

- W F d - m N d - m mg d DK 0 ½ mv2
- - m m1g d1 - m m2g d2 ? d1 / d2 m2 / m1

(A) m1 (B) m2 (C) They will go the same

distance

m1

v1

v2

m2

Home Exercise Work/Energy for Non-Conservative

Forces

- The air track is once again at an angle of 30

with respect to horizontal. The cart (with mass 1

kg) is released 1 meter from the bottom and hits

the bumper at a speed, v1. This time the vacuum/

air generator breaks half-way through and the air

stops. The cart only bounces up half as high as

where it started. - How much work did friction do on the cart ?(g10

m/s2)

1 meter

30

(A) 2.5 J (B) 5 J (C) 10 J (D) 2.5 J (E)

5 J (F) 10 J

Home Exercise Work/Energy for Non-Conservative

Forces

- How much work did friction do on the cart ? (g10

m/s2) - W F Dx is not easy to do
- Work done is equal to the change in the energy of

the system (U and/or K). Efinal - Einitial and

is lt 0. (E UK) - Use W Ufinal - Uinit mg ( hf - hi ) - mg

sin 30 0.5 m - W -2.5 N m -2.5 J or (D)

hi

hf

1 meter

30

(A) 2.5 J (B) 5 J (C) 10 J (D) 2.5 J (E)

5 J (F) 10 J

Non-conservative Forces

- If the work done does not depend on the path

taken, the force involved is said to be

conservative. - If the work done does depend on the path taken,

the force involved is said to be

non-conservative. - An example of a non-conservative force is

friction - Pushing a box across the floor, the amount of

work that is done by friction depends on the path

taken. - and work done is proportional to the length of

the path !

A Non-Conservative Force, Friction

- Looking down on an air-hockey table with no air

flowing (m gt 0). - Now compare two paths in which the puck starts

out with the same speed (Ki path 1 Ki path 2) .

A Non-Conservative Force

Since path2 distance gtpath1 distance the puck

will be traveling slower at the end of path 2.

Work done by a non-conservative force

irreversibly removes energy out of the system.

Here WNC Efinal - Einitial lt 0 ? and

reflects Ethermal

Work Power

- Two cars go up a hill, a Corvette and a ordinary

Chevy Malibu. Both cars have the same mass. - Assuming identical friction, both engines do the

same amount of work to get up the hill. - Are the cars essentially the same ?
- NO. The Corvette can get up the hill quicker
- It has a more powerful engine.

Work Power

- Power is the rate at which work is done.
- Average Power is,
- Instantaneous Power is,
- If force constant, W F Dx F (v0 Dt ½ aDt2)
- and P W / Dt F (v0 aDt)

Work Power

- Power is the rate at which work is done.

Units (SI) are Watts (W)

Instantaneous Power

Average Power

1 W 1 J / 1s

Example

- A person of mass 80.0 kg walks up to 3rd floor

(12.0m). If he/she climbs in 20.0 sec what is

the average power used. - Pavg F h / t mgh / t 80.0 x 9.80 x 12.0 /

20.0 W - P 470. W

Exercise Work Power

- Starting from rest, a car drives up a hill at

constant acceleration and then suddenly stops at

the top. - The instantaneous power delivered by the engine

during this drive looks like which of the

following,

- Top
- Middle
- Bottom

Chap. 12 Rotational Dynamics

- Up until now rotation has been only in terms of

circular motion with ac v2 / R and aT d

v / dt - Rotation is common in the world around us.
- Many ideas developed for translational motion are

transferable.

Conservation of angular momentum has consequences

How does one describe rotation (magnitude and

direction)?

Rotational Dynamics A childs toy, a physics

playground or a students nightmare

- A merry-go-round is spinning and we run and jump

on it. What does it do? - We are standing on the rim and our friends spin

it faster. What happens to us? - We are standing on the rim a walk towards the

center. Does anything change?

Rotational Variables

- Rotation about a fixed axis
- Consider a disk rotating about an axis through

its center - How do we describe the motion
- (Analogous to the linear case )

Rotational Variables...

- Recall At a point a distance R away from the

axis of rotation, the tangential motion - x ? R
- v ? R
- a ? R

Overview (with comparison to 1-D kinematics)

- Angular Linear

And for a point at a distance R from the rotation

axis

x R ????????????v ? R ??????????aT ? R

Here aT refers to tangential acceleration

Exercise Rotational Definitions

- A friend at a party (perhaps a little tipsy) sees

a disk spinning and says Ooh, look! Theres a

wheel with a negative w and positive a! - Which of the following is a true statement about

the wheel?

- The wheel is spinning counter-clockwise and

slowing down. - The wheel is spinning counter-clockwise and

speeding up. - The wheel is spinning clockwise and slowing down.
- The wheel is spinning clockwise and speeding up

Lecture 15

- Assignment
- HW7 due March 25th
- For Thursday Read Chapter 12, Sections 7-11
- do not concern yourself with the integration

process in regards to center of mass or moment

of inertia