A) > B) < C) =

Example 6.9g

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 B) < C) =

Example 6.9h

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 B) < C) =

Example 6.9i

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 >M 1 collision is perfectly elastic. and which is initially at rest. The Just after the collision v 2 ______ v 0 .

A) > B) < C) =

Example 6.9j

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 >M 1 collision is perfectly elastic. and which is initially at rest. The Just after the collision v 1 ______ 0.

A) > B) < C) =

Example 6.9k

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 >M 1 collision is perfectly elastic. and which is initially at rest. The Just after the collision P 2 ______ M 1 v 0 . A) > B) < C) =

Example 6.9l

The mass M 1 enters from the left with velocity v 0 strikes the mass M 2 >M 1 collision is perfectly elastic. and which is initially at rest. The At maximum compression, the energy stored in the spring is ________ (1/2)M 1 v 0 2 A) > B) < C) =

Example 6.10

Ballistic Pendulum: used to measure speed of bullet. 0.5-kg block of wood swings up by height h = 65 cm after stopping 8.0-g bullet.

227 m/s What was bullet’s velocity?

Example 6.11

A 5-g bullet traveling at 500 m/s embeds in a 1.495 kg block of wood resting on the edge of a 0.9-m high table. How far does the block land from the edge of the table?

71.4 cm

Example 6.12

Tarzan (M=80 kg) swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical. At the bottom of his swing, he picks up Jane (m=50 kg). To what angle do Tarzan and Jane swing?

35.8 degrees (indepedent of L or g)

Example 6.12a

Tarzan (M=80 kg) swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical. At the bottom of his swing, he picks up Jane (m=50 kg). To what angle do Tarzan and Jane swing?

To calculate Tarzan’s speed just before he picks up Jane, you should apply: A) Conservation of Energy B) Conservation of Momentum

Example 6.12b

Tarzan (M=80 kg) swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical. At the bottom of his swing, he picks up Jane (m=50 kg). To what angle do Tarzan and Jane swing?

To calculate Tarzan&Jane’s speed just after their collision (given the previous answer) you should apply: A) Conservation of Energy B) Conservation of Momentum

Example 6.12c

Tarzan (M=80 kg) swings on a 12 m vine by letting go from an angle of 60 degrees from the vertical. At the bottom of his swing, he picks up Jane (m=50 kg). To what angle do Tarzan and Jane swing?

To calculate Tarzan&Jane’s final height (given the previous answer) you should apply: A) Conservation of Energy B) Conservation of Momentum

Chapter 7

Rotational Motion

Universal Law of Gravitation Kepler’s Laws

• •

Angular Displacement

•

Circular motion about a fixed AXIS Use polar coordinates: (r,



) Distance r doesn’t change

• • • •

Three possible units for



: Revolutions Degrees (1 rev. = 360 deg.) Radians (1 rev. = 2

p

rad.)



Angular Displacement, cont.

•

Distance traveled by a point: (distance r from axis)

s

  2 p

r N r

 (N counts revolutions) (  is in radians)  (rad)  2 p

N

(revolutions)

Example 7.1

An automobile wheel has a radius of 42 cm. If a car drives 10 km, through what angle has the wheel rotated?

a) In revolutions b) In radians c) In degrees a) N = 3789 b)



= 2.38x10

4 radians c)



= 1.36x10

6 degrees



Angular Speed

    

t

 

f t f

 

i



t i

•

(in rad/s) Can also be given in

•

Revolutions/s

•

Degrees/s

 •

Linear (tangential) Speed at r

v t

 

s



t



r

  

t v t



r



(



in rad/s)



Example 7.2

A race car engine can turn at a maximum rate of 12,000 rpm. (revolutions per minute).

a) What is the angular velocity in radians per second.

b) If helicopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound. DATA: The speed of sound is 343 m/s a) 1256 rad/s b) 27 cm

Angular Acceleration

• • • •

Denoted by

a a  

f

 

i t

 a

in rad/s rad/s² Every point on rigid object has same



and

a

Rotational/Linear Correspondence:

   0 

f

a

t

 

x



v

0 

v f



a



t

Rotational/Linear Correspondence, cont’d

Rotational Motion

     0  2 

f



t

Linear Motion



v

0 

x

  2

v f



t



f

  0  a

t v f



v

0 

at

    0

t

 1 2 a

t

2    

f t

 1 2 a

t

2  2

f

 a   2   0 2 2 

x



v

0

t

 1 2

at

2 

x



v f t

 1 2

at

2

v f

2 2 

v

0 2 2 

a



x

Example 7.3

A pottery wheel is accelerated uniformly from rest to a rotation speed of 10 rpm in 30 seconds.

a.) What was the angular acceleration? (in rad/s 2 ) b.) How many revolutions did the wheel undergo during that time?

a) 0.0349 rad/s 2 b) 2.50 revolutions

Linear movement of a rotating point

• • •

Distance

x



Speed

v



r



r

 

Acceleration

a



r

a

Angles must be in radians!

Different points have different linear speeds!

Special Case - Rolling Motion

• •

Wheel (radius r) rolls without slipping Angular motion of wheel gives linear motion of car

• • •

Distance

x



r

 

Speed

v



r



Acceleration

a



r

a

Example 7.4

A coin of radius 1.5 cm is initially rolling with a rotational speed of 3.0 radians per second, and comes to a rest after experiencing a slowing down of

a

= 0.05 rad/s 2 .

a.) Over what angle (in radians) did the coin rotate?

b.) What linear distance did the coin move?

a) 90 rad b) 135 cm