1. PHYSICS 231 INTRODUCTORY PHYSICS I
Lecture 9

2. Last Lecture Momentum: Newton’s 2nd: Conserved for isolated objects
Useful for short times
Collisions (p always conserved)
Elastic (E conserved too)
Inelastic (E not conserved)

3. Example 6.7
A proton (mp=1.67x10-27 kg) elastically collides with a target proton which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target proton bounces forward, what are
a) the final velocity of the projectile proton?
b) the final velocity of the target proton?
0.0
3.0x106 m/s

4. Elastic collision in 1-dimension
Conservation of Energy:
Conservation of Momentum:
Rearrange both equations and divide:

5. Elastic collision in 1-dimension
Final equations for head-on elastic collision:
Relative velocity changes sign
Equivalent to Conservation of Energy

6. Example 6.8
An proton (mp=1.67x10-27 kg) elastically collides with a target deuteron (mD=2mp) which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target deuteron bounces forward, what are
a) the final velocity of the projectile proton?
b) the final velocity of the target deuteron?
vp =-1.0x106 m/s
vd = 2.0x106 m/s
Head-on collisions with heavier objects always lead to reflections

7. Example 6.9a The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The collision is perfectly elastic.
a) Just after the collision v2 ______ v0.
A) >
B) <
C) =

8. Example 6.9b The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =

9. Example 6.9c The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =

10. Example 6.9d The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The collision is perfectly elastic.
At maximum compression, the energy stored in the spring is ________ (1/2)M1v02
A) >
B) <
C) =

11. Example 6.9e The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision v2 ______ v0.
A) >
B) <
C) =

12. Example 6.9f The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =

13. Example 6.9g The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =

14. Example 6.9h The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The collision is perfectly elastic.
At maximum compression, the energy stored in the spring is ________ (1/2)M1v02
A) >
B) <
C) =

15. Example 6.9i The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision v2 ______ v0.
A) >
B) <
C) =

16. Example 6.9j The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =

17. Example 6.9k The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =

18. Example 6.9l The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The collision is perfectly elastic.
At maximum compression, the energy stored in the spring is ________ (1/2)M1v02
A) >
B) <
C) =

19. 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. What was bullet’s velocity?
227 m/s

20. Example 6.11
A 5-g bullet traveling at 500 m/s embeds in a 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

21. 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)

22. 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

23. 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

24. 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

25. Rotational Motion Universal Law of Gravitation Kepler’s Laws
Chapter 7
Rotational Motion Universal Law of Gravitation Kepler’s Laws

26. 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. = 2p rad.)

27. Angular Displacement, cont.
Distance traveled by a point:
(distance r from axis)

28. 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) q = 2.38x104 radians
c) q = 1.36x106 degrees

29. Angular Speed Linear (tangential) Speed at r (in rad/s)
Can also be given in
Revolutions/s
Degrees/s
Linear (tangential) Speed at r
( in rad/s)

30. 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

31. Angular Acceleration Denoted by a w in rad/s  rad/s²
Every point on rigid object has same w and a

32. Rotational/Linear Correspondence:

33. Rotational/Linear Correspondence, cont’d
Rotational Motion
Linear Motion
Constant 
Constant a

34. 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/s2)
b.) How many revolutions did the wheel undergo during that time?
a) rad/s2
b) 2.50 revolutions

35. Linear movement of a rotating point
Distance
Speed
Acceleration
Different points have different linear speeds!
Angles must be in radians!

36. Special Case - Rolling Motion
Wheel (radius r) rolls without slipping
Angular motion of wheel gives linear motion of car
Distance
Speed
Acceleration

37. 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/s2.
a.) Over what angle (in radians) did the coin rotate?
b.) What linear distance did the coin move?
a) 90 rad
b) 135 cm