Exam 3: 2103 Chamberlin Hall, B102 Van Vleck

1
Lecture 23

• Exam 3 2103 Chamberlin Hall, B102 Van Vleck
  quiet room
• CH Sections 604 605 606 609 610 611 (TA Moriah
  T.,  Abdallah C., Tamara S.)
• VV Sections 602 603 607 608 612 (TA Eric P., Dan
  C., Zhe D.)
• Format Closed book, three 8 x11 sheets, hand
  written
• Electronics Any calculator is okay but no
  web/cell access
• Quiet room Test anxiety, special accommodations,
  etc.
• Chapters Covered
• Chapter 9 Linear momentum and collision (not
  9.9)
• Chapter 10 Rotation about fixed axis and rolling
• Chapter 11 Angular Momentum (not 11.5)
• Chapter 12 Static equilibrium and elasticity

2
Basic Concepts and Quantities

• Momentum, Angular Momentum
• Linear Momentum, Impulse
• Angular Momentum (magnitude and direction)
• Torque
• Collisions Elastic Inelastic
• Center of Mass
• Rotational Motion (1-axis)
• Angular displacement (?q) / Velocity(w),
  Acceleration (a).
• Moments of Inertia
• Rotational Kinetic Energy
• Conservation Laws Energy, Momentum, Angular
  Momentum
• Static Equilibrium
• Elasticity Youngs, Shear, Bulk Modulus

3
Chapter 9 Momentum and Momentum Conservation
Ix
Ix
4
Chapter 9
5
Chapter 9

• Center of Mass

6
Chapter 10

• At a point a distance R away from the axis of
  rotation, the tangential motion
• s (arc) ? R
• vT (tangential) ? R
• aT ? R

7
Chapter 10
8
Chapter 10
9
Chapter 11
10
Chapter 12
11
Approach to Statics

• In general, we can use the two equations
• to solve any statics problems.
• When choosing axes about which to calculate
  torque, choose one that makes the problem
  easy....
• However if there is acceleration, then restrict
  rotation axis to center of mass (as well as for
  translation).

12
Changes in length Youngs modulus

• Youngs modulus measures the resistance of a
  solid to a change in its length.

F
elasticity in length
A
L0
?L
13
Changes in volume Bulk Modulus

• Bulk modulus measures the resistance of solids
  or liquids to changes in their volume.

Volume elasticity
14
Changes in shape Shear Modulus
Applying a force perpendicular to a surface
15
Comparison Kinematics

• Angular Linear

16
Comparison Dynamics

• Angular Linear

m
I Si mi ri2
F a m
t r x F a I
L r x t I w
p mv
W F ?x
W ? D?
?K WNET
?K WNET
17
Momentum and collisions

• Remember vector components
• A 5 kg cart rolling without friction to the right
  at 10 m/s collides and sticks to a 5 kg
  motionless block on a 30 frictionless incline.
• How far along the incline do the joined blocks
  slide?

18
Momentum and collisions

• Remember vector components
• A 5 kg cart rolling without friction to the right
  at 10 m/s collides and sticks to a 5 kg
  motionless block on a q30 frictionless incline.
  
• Momentum parallel to incline is conserved
• Normal force (by ground on cart) is ? to the
  incline
• mvi cos q m 0 2m vf ? vf vi cos q /
  2 4.4 m/s
• Now use work-energy
• 2mgh 0 ½ 2m v2f ? d h / sin q v2f
  / (2g sin q)

mvi cos q
mvi

19
Example problem Going in circles A

• A 2.0 kg disk tied to a 0.50 m string undergoes
  circular motion on a rough but horizontal table
  top. The kinetic coefficient of friction is
  0.25. If the disk starts out at 5.0 rev/sec how
  many revolutions does it make before it comes to
  rest?
• Work-energy theorem
• W F d 0 ½ mv2
• F -mmg d - ½ mv2
• d v2 /(2mg)(5.0 x 2p x 0.50)2/ (0.50 x 10) m
  5 p2 m
• Rev d / 2pr 16 revolutions
• What if the disk were tilted by 60 ?

Top view
20
Example problem Going in circles B

• A 2.0 kg disk tied to a long string undergoes
  circular motion on a frictionless horizontal
  table top. The string passes through a hole and
  then hangs vertically. The disk starts out at 5.0
  rev/sec 0.50 m away from hole. If you pull
  slowly down on the string so that the final
  radius is 0.25 m, what is the final angular
  velocity?
• No external torque so angular momentum is
  conserved.
• Ib wb Ia wa
• I mR2
• (0.50)2 wb (0.25)2 wa
• 4 wb wa 20 rev/sec

21
Spinning ball on incline

• A solid disk of mass m and radius R is spinning
  with angular velocity w. It is positioned so that
  it can either move directly up or down an incline
  of angle q (but it is not rolling motion). The
  coefficient of kinetic friction is m. At what
  angle will the disks position on the incline not
  change?

w
q
22
Spinning ball on incline

• A solid disk of mass m and radius R is spinning
  with angular velocity w. It is positioned so that
  it can either move directly up or down an incline
  of angle q (but it is not rolling motion). The
  coefficient of kinetic friction is m.
• At what angle will the disks position on the
  incline not change?

23
Spinning ball on incline

• A solid disk of mass m and radius r is spinning
  with angular velocity w. It is positioned so that
  it can either move directly up or down an incline
  of angle q (but it is not rolling motion). The
  coefficient of kinetic friction is m. While
  spinning the disks position will not change.
• How long will it be before it starts to roll?
• This will occur only when w0.

24
Example Problem

• A 3.0 kg mass is attached to a light, rigid rod
  1.5 m long. The rod is vertical and is anchored
  to the ground through a frictionless pivot. It
  sits perfectly balanced at an unstable
  equilibrium. A 500 gm bullet is shot
  horizontally at 100 m/s through the mass. The
  force versus time plot is shown.
• How fast is the bullet going when
• it leaves?
• What is the tension in the rod just
• after the bullet exits?

Method Impulse
I area under curve
25
Example Problem

• A 3.0 kg mass is attached to a light, rigid rod
  1.5 m long. The rod is vertical and is anchored
  to the ground through a frictionless pivot. It
  sits perfectly balanced at an unstable
  equilibrium. A 500 gm bullet is shot
  horizontally at 100 m/s through the mass. The
  force versus time plot is shown.
• How fast is the bullet going when
• it leaves?

26
Example Problem

• A 3.0 kg mass is attached to a light, rigid rod
  1.5 m long. The rod is vertical and is anchored
  to the ground through a frictionless pivot. It
  sits perfectly balanced at an unstable
  equilibrium. A 500 gm bullet is shot
  horizontally at 100 m/s through the mass. The
  force versus time plot is shown.
• What is the tension in the rod just
• after the bullet exits?

27
Statics Example

• A sign of mass M is hung 1 m from the end of a
  4 m long beam (mass m) as shown in the diagram.
  The beam is hinged at the wall. What is the
  tension in the wire in terms of m, M, g and any
  other given quantity?

wire
q 30o
1 m
SIGN
28
Statics Example
T
Fy

30
X
Fx
mg
2 m
Mg
3 m
Process Make a FBD and note known / unknown
forces. Chose axis of rotation at support because
Fx Fy are not known

• F 0 ? 0 Fx T cos 30
• 0 Fy T sin 30 - mg - Mg
• z-dir Stz 0 -mg 2r Mg 3r T sin 30 4r
  (r 1m)
• The torque equation get us where we need to go, T
• T (2m 3M) g / 2

29
Center of Mass Example Astronauts Rope

• Two astronauts are initially at rest in outer
  space and 20 meters apart. The one on the right
  has 1.5 times the mass of the other (as shown).
  The 1.5 m astronaut wants to get back to the ship
  but his jet pack is broken. There happens to be
  a rope connected between the two. The heavier
  astronaut starts pulling in the rope.
• (1) Does he/she get back to the ship ?
• (2) Does he/she meet the other astronaut ?

M 1.5m
m
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Example Astronauts Rope

1. There is no external force so if the larger
   astronaut pulls on the rope he will create an
   impulse that accelerates him/her to the left and
   the small astronaut to the right. The larger
   ones velocity will be less than the smaller
   ones so he/she doesnt let go of the rope they
   will either collide (elastically or
   inelastically) and thus never make it.

31
Example Astronauts Rope

• (2) However if the larger astronaut lets go of
  the rope he will get to the ship. (Too bad for
  the smaller astronaut!)
• In all cases the center of mass will remain fixed
  because they are initially at rest and there is
  no external force.
• To find the position where they meet all we need
  do is find the Center of Mass

32
Forces and rigid body rotation

• To change the angular velocity of a rotating
  object, a force must be applied
• How effective an applied force is at changing the
  rotation depends on several factors
• The magnitude of the force
• Where, relative to the axis of rotation the force
  is applied
• The direction of the force

A
B
C
Which applied force will cause the wheel to spin
the fastest?
33
Leverage

• The same concept applies to leverage
• the lever undergoes rigid body rotation about a
  pivot point

B
C
A
Which applied force provides the greatest lift ?
34
More on torques

• You need to change the tire on you car. You use a
  tire wrench which allows you to apply a pair of
  forces.
• (A) What is the torque produced by a tire wrench
  of length L, given an applied couple of magnitude
  F, acting on a lug nut (point F) as shown in the
  figure?
• (B) Assume the lug nut is stuck What is the
  torque acting on the wheel, if the lug nut is a
  distance r from the center?

Image courtesy John Wiley Sons, Inc.
35
Wheel wrench

• 1. tF (L/2) F (L/2) F LF
• 2. tF r F sinf r F sin ( p-q )
• LF
• 3. tF L F 0 F
• Notice the torque is the same everywhere.

f
f
36
For Thursday

• Chapter 13 (Newtons Law of Gravitation)

37
Momentum Impulse

• A rubber ball collides head on (i.e., velocities
  are opposite) with a clay ball of the same mass.
  The balls have the same speed, v, before the
  collision, and stick together after the
  collision. What is their speed immediately after
  the collision?

1. 0
2. ½ v
3. 2 v
4. 4 v

38
Momentum Impulse

• A rubber ball collides head on with a clay ball
  of the same mass. The balls have the same speed,
  v, before the collision, and stick together after
  the collision. What is their speed after the
  collision?
• (a) 0
• (b) ½ v
• (c) 2 v
• (d) 4 v

39
Momentum, Work and Energy

• A 0.40 kg block is pushed up against a spring
  (with spring constant 270 N/m ) on a frictionless
  surface so that the spring is compressed 0.20 m.
  When the block is released, it slides across the
  surface and collides with the 0.60 kg bob of a
  pendulum. The bob is made of clay and the block
  sticks to it. The length of the pendulum is 0.80
  m. (See the diagram.)
• To what maximum height above the surface will the
  ball/block assembly rise after the collision?
  (g9.8 m/s2)
• A. 2.2 cm
• B. 4.4 cm
• C. 11. cm
• D. 22 cm
• E. 44 cm
• F. 55 cm

40
Momentum, Work and Energy

• A 0.40 kg block is pushed up against a spring
  (with spring constant 270 N/m ) on a frictionless
  surface so that the spring is compressed 0.20 m.
  When the block is released, it slides across the
  surface and collides with the 0.60 kg bob of a
  pendulum. The bob is made of clay and the block
  sticks to it. The length of the pendulum is .80
  m. (See the diagram.)
• To what maximum height above the surface will the
  ball/block assembly rise after the collision?
• A. 2.2 cm
• B. 4.4 cm
• C. 11. cm
• D. 22 cm
• E. 44 cm
• F. 55 cm

41
Momentum, Work and Energy ( Now with friction)

• A 0.40 kg block is pushed up against a spring
  (with spring constant 270 N/m ) on a surface.
• If mstatic 0.54, how far can the spring be
  compressed and the block remain stationary (i.e.,
  maximum static friction)?
• S F 0 k u - f k u - m N
• u m mg/k 0.54 (0.40x10 N) / 270 N/m
  0.0080 m

42
Momentum, Work and Energy ( Now with friction)

• A 0.40 kg block is pushed up against a spring
  (with spring constant 270 N/m ) on a surface.
  The spring is compressed 0.20 m
• If mkinetic 0.50 and the block is 9.8 m away
  from the unstretched spring, how high with the
  clay/block pair rise?
• Emech (at collision) Uspring Wfriction ½ k
  u2 - m mg d
• 1/2 m v2 135(0.04)-0.50(0.40x10.)10.(540-20)
  J520 J
• v2 1040/0.40 m2/s2
• Now the collision (cons. of momentum) and the
  swing.

43
Momentum and Impulse

• Henri Lamothe holds the world record for the
  highest shallow dive. He belly-flopped from a
  platform 12.0 m high into a tank of water just
  30.0 cm deep! Assuming that he had a mass of 50.0
  kg and that he stopped just as he reached the
  bottom of the tank, what is the magnitude of the
  impulse imparted to him while in the tank of
  water (in units of kg m/s)?
• (a) 121
• (b) 286
• (c) 490
• (d) 623
• (e) 767

44
Momentum and Impulse

• Henri Lamothe holds the world record for the
  highest shallow dive. He belly-flopped from a
  platform 12.0 m high into a tank of water just
  30.0 cm deep! Assuming that he had a mass of 50.0
  kg and that he stopped just as he reached the
  bottom of the tank, what is the magnitude of the
  impulse imparted to him while in the tank of
  water (in units of kg m/s)?
• (a) 121
• (b) 286
• (c) 490
• (d) 623
• (e) 767

Dp sqrt(2x9.8x12.3)x50
45
Momentum Impulse

• Suppose that in the previous problem, the
  positively charged particle is a proton and the
  negatively charged particle, an electron. (The
  mass of a proton is approximately 1,840 times the
  mass of an electron.) Suppose that they are
  released from rest simultaneously. If, after a
  certain time, the change in momentum of the
  proton is Dp, what is the magnitude of the change
  in momentum of the electron?
• (a) Dp / 1840
• (b) Dp
• (c) 1840 Dp

46
Momentum Impulse

• Suppose that in the previous problem, the
  positively charged particle is a proton and the
  negatively charged particle, an electron. (The
  mass of a proton is approximately 1,840 times the
  mass of an electron.) Suppose that they are
  released from rest simultaneously. If, after a
  certain time, the change in momentum of the
  proton is Dp, what is the magnitude of the change
  in momentum of the electron?
• (a) Dp / 1840
• (b) Dp
• (c) 1840 Dp

47
Conservation of Momentum

• A woman is skating to the right with a speed of
  12.0 m/s when she is hit in the stomach by a
  giant snowball moving to the left. The mass of
  the snowball is 2.00 kg, its speed is 25.0 m/s
  and it sticks to the woman's stomach. If the mass
  of the woman is 60.0 kg, what is her speed after
  the collision?
• (a) 10.8 m/s
• (b) 11.2 m/s
• (c) 12.4 m/s
• (d) 12.8 m/s

48
Conservation of Momentum

• A woman is skating to the right with a speed of
  12.0 m/s when she is hit in the stomach by a
  giant snowball moving to the left. The mass of
  the snowball is 2.00 kg, its speed is 25.0 m/s
  and it sticks to the woman's stomach. If the mass
  of the woman is 60.0 kg, what is her speed after
  the collision?
• (a) 10.8 m/s
• (b) 11.2 m/s
• (c) 12.4 m/s
• (d) 12.8 m/s

49
Conservation of Momentum

• Sean is carrying 24 bottles of beer when he slips
  in a large frictionless puddle. He slides
  forwards with a speed of 2.50 m/s towards a very
  steep cliff. The only way for Sean to stop before
  he reaches the edge of the cliff is to throw the
  bottles forward at 20.0 m/s (relative to the
  ground). If the mass of each bottle is 500 g, and
  Sean's mass is 72 kg, what is the minimum number
  of bottles that he needs to throw?
• (a) 18
• (b) 20
• (c) 21
• (d) 24
• (e) more than 24

50
Momentum and Impulse

• A stunt man jumps from the roof of a tall
  building, but no injury occurs because the person
  lands on a large, air-filled bag. Which one of
  the following statements best describes why no
  injury occurs?
• (a) The bag provides the necessary force to stop
  the person.
• (b) The bag reduces the impulse to the person.
• (c) The bag reduces the change in momentum.
• (d) The bag decreases the amount of time during
  which the momentum is changing and reduces the
  average force on the person.
• (e) The bag increases the amount of time during
  which the momentum is changing and reduces the
  average force on the person.

51
Momentum and Impulse

• A stunt man jumps from the roof of a tall
  building, but no injury occurs because the person
  lands on a large, air-filled bag. Which one of
  the following statements best describes why no
  injury occurs?
• (a) The bag provides the necessary force to stop
  the person.
• (b) The bag reduces the impulse to the person.
• (c) The bag reduces the change in momentum.
• (d) The bag decreases the amount of time during
  which the momentum is changing and reduces the
  average force on the person.
• (e) The bag increases the amount of time during
  which the momentum is changing and reduces the
  average force on the person.

52
Momentum and Impulse

• Two blocks of mass m1 M and m2 2M are both
  sliding towards you on a frictionless surface.
  The linear momentum of block 1 is half the linear
  momentum of block 2. You apply the same constant
  force to both objects in order to bring them to
  rest. What is the ratio of the two stopping
  distances d2/d1?
• (a) 1/ 2
• (b) 1/ 2½
• (c) 1
• (d) 2½
• (e) 2
• (f) Cannot be determined without knowing the
  masses of the objects and their velocities.

53
Momentum and Impulse

• Two blocks of mass m1 M and m2 2M are both
  sliding towards you on a frictionless surface.
  The linear momentum of block 1 is half the linear
  momentum of block 2. You apply the same constant
  force to both objects in order to bring them to
  rest. What is the ratio of the two stopping
  distances d2/d1?
• (a) 1/ 2
• (b) 1/ 2½
• (c) 1
• (d) 2½
• (e) 2
• (f) Cannot be determined without knowing the
  masses of the objects and their velocities.

54
Momentum and Impulse

• In a table-top shuffleboard game, a heavy moving
  puck collides with a lighter stationary puck as
  shown. The incident puck is deflected through an
  angle of 20 and both pucks are eventually
  brought to rest by friction with the table. The
  impulse approximation is valid (i.e.,the time of
  the collision is short relative to the time of
  motion so that momentum is conserved).
• Which of the following
• statements is correct?
• A. The collision must be inelastic because the
  pucks have different masses.
• B. The collision must be inelastic because there
  is friction between the pucks and the surface.
• C. The collision must be elastic because the
  pucks bounce off each other.
• D. The collision must be elastic because, in the
  impulse approximation,
• momentum is conserved.
• E. There is not enough information given to
  decide whether the collision is
• elastic or inelastic.

55
Momentum and Impulse

• In a table-top shuffleboard game, a heavy moving
  puck collides with a lighter stationary puck as
  shown. The incident puck is deflected through an
  angle of 20 and both pucks are eventually
  brought to rest by friction with the table. The
  impulse approximation is valid (i.e.,the time of
  the collision is short relative to the time of
  motion so that momentum is conserved).
• Which of the following
• statements is correct?
• A. The collision must be inelastic because the
  pucks have different masses.
• B. The collision must be inelastic because there
  is friction between the pucks and the surface.
• C. The collision must be elastic because the
  pucks bounce off each other.
• D. The collision must be elastic because, in the
  impulse approximation,
• momentum is conserved.
• E. There is not enough information given to
  decide whether the collision is
• elastic or inelastic.

56
Exercise Ladder against smooth wall

• Bill (mass M) is climbing a ladder (length L,
  mass m) that leans against a smooth wall (no
  friction between wall and ladder). A frictional
  force F between the ladder and the floor keeps it
  from slipping. The angle between the ladder and
  the wall is ?.
• What is the magnitude of F as a function of
  Bills distance up the ladder?

?
L
m
Bill
F
57
Ladder against smooth wall...

• Consider all of the forces acting. In addition to
  gravity and friction, there will be normal forces
  Nf and Nw by the floor and wall respectively on
  the ladder.
• First sketch the FBD

Nw
L/2
?

• Again use the fact that FNET 0
  in both x and y directions
• x Nw F
• y Nf Mg mg

mg
d
Mg
F
Nf
58
Ladder against smooth wall...

• Since we are not interested in Nw, calculate
  torques about an axis through the top end of the
  ladder, in the z direction.

torque axis
Nw
?
L/2
m

• Substituting Nf Mg mg andsolve for F

mg
a
d
Mg
F
Nf
a
59
Example Ladder against smooth wall
We have just calculated that

• For a given coefficient of static friction
  ?s,the maximum force of friction F that can
  beprovided is ?sNf ?s g(M m).
• The ladder will slip if F exceedsthis value.

?
m
Cautionary note (1) Brace the bottom of
ladders! (2) Dont make ? too big!
d
F