Physics 215 – Fall 2014 Lecture 06-2 1

Welcome back to Physics 215

Today’s agenda:

• Friction, drag• Tension, pulleys• Forces in circular motion• Impulse

Physics 215 – Fall 2014 Lecture 06-2 2

Current homework assignment

• HW5:– Knight textbook Ch.6: 38, 42, 56, 58– Ch.7: 46, 54– due Friday, Oct. 3rd in recitation

Physics 215 – Fall 2014 Lecture 06-2 3

Summary of friction

• 2 laws of friction: static and kinetic

• Static friction tends to oppose motion and is governed by inequality

Fs ≤ sN

• Kinetic friction is given by equality FK = KN

Physics 215 – Fall 2014 Lecture 06-2 4

The magnitude of the force of kinetic friction between two objects

• depends on the type of surfaces of the objects• depends on the normal force that the objects exert on each

other• does not depend on the surface area where the two objects are

touching • does not depend on the speed with which one object is moving

relative to the other

What if > tan-1s ?

Physics 215 – Fall 2014 Lecture 06-2 5

What if > tan-1s ?

• Block begins to slide

• Resolve along plane:

Wsin- Wcos= ma

• Or:

a = g(sin- cos)

Physics 215 – Fall 2014 Lecture 06-2 6

Tension

• For an ideal string or rope connecting two

objects:

• does not stretch inextensible

• has zero mass

• Let’s look at an example of a cart connected to a falling mass by an ideal string...

Physics 215 – Fall 2014 Lecture 06-2 7

1. equal to 1.0 N (i.e. the weight of A),

2. between 1.0 N and 1.5 N,

3. equal to 1.5 N (i.e. the weight of C), or

4. equal to 2.5 N (i.e. the sum of their weights)?

Blocks A and C are initially held in place as shown. After the blocks are released, block A will accelerate up and block C will accelerate down.

The magnitudes of their accelerations are the same.

Will the tension in the string be

Physics 215 – Fall 2014 Lecture 06-2 8

Demo: Pulleys

F

W

T

*2 pulleys2T = W

F = T = W/2

*N pulleysF = W/N!

Physics 215 – Fall 2014 Lecture 06-2 9

Forces in circular motion

• Motion around circular track, constant speed (for now):

arad = v2/r

Physics 215 – Fall 2014 Lecture 06-2 10

1. four times as great

2. twice as great

3. equal to

4. one half as great

as the acceleration of ball A.

Two identical balls are connected by a string and whirled around in circles of radius r and 2r at angular speed.

The acceleration of ball B is

Physics 215 – Fall 2014 Lecture 06-2 11

1. less than

2. equal to

3. greater than

the tension in string R.

The two identical balls are whirled around in a circle as before. Assume that the balls are moving very fast and that the two strings are massless.

The tension in string P is

Physics 215 – Fall 2014 Lecture 06-2 12

Car on circular track with friction

• Motion around circular track, constant speed (for now):

arad = v2/r

Physics 215 – Fall 2014 Lecture 06-2 13

Motion of car on banked circular track (no friction)

R

Speed v

carN

W

Horizontal forces:

Vertical:

a =

Physics 215 – Fall 2014 Lecture 06-2 14

Motion on loop-the-loop

car

What is normal force on car at top and bottom of loop?Neglect friction; assume moves with speed vB at bottom and vT at top

At bottom At top

Physics 215 – Fall 2014 Lecture 06-2 15

Impulse

• Constant force F12 acting on object 1 due to object 2 for a time t yields an impulse

I12 = F12 t • In general, for a time varying force need to

use this for small t and add:

I = F(t) t =

Physics 215 – Fall 2014 Lecture 06-2 16

Impulse for time varying forces

* area under curve equals impulse

t

F(t)

t1 t2

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Impulse change in momentum

• Consider first constant forces ...

• Constant acceleration equation:

vf = vi + at

mvf - mvi = mat =

• If we call p = mv momentum we see that

p =

Physics 215 – Fall 2014 Lecture 06-2 18

Impulse demo

• Cart equipped with force probe collides with rubber tube

• Measure force vs. time and momentum vs. time

• Find that integral of force curve is precisely the change in p!

Physics 215 – Fall 2014 Lecture 06-2 19

Momentum of an object:

p = mv

Definitions of impulse and momentum

Impulse imparted to object 1 by object 2:

I12 = F12 t

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The net impulse imparted to an object is equal to its change in its momentum.

Impulse-momentum theorem

Inet = p

Physics 215 – Fall 2014 Lecture 06-2 21

1. Case A.

2. Case B.

3. Case C.

4. Cases A and B.

Consider the change in momentum in these three cases:

A. A ball moving with speed v is brought to rest.

B. The same ball is projected from rest so that it moves with speed v.

C. The same ball moving with speed v is brought to rest and immediately projected backward with speed v.

In which case(s) does the ball undergo the largest change in momentum?

Physics 215 – Fall 2014 Lecture 06-2 22

Reading assignment

• Momentum, collisions

• Chapter 9 in textbook