Agenda:

1
Lecture 16

• Agenda
• Review for exam
• Assignment For Tuesday, Read chapter 10.1-10.5
• M. Tobins office hours today are from 305 to
  355 PM
• I will have extra office hours tomorrow from 1 to
  230 PM

2
Newtons Laws

• Three blocks are connected on the table as shown.
  
• The table has a coefficient of kinetic friction
  of 0.350, the masses are m1 4.00 kg, m2
  1.00kg and m3 2.00kg.

• If m3 starts from rest how fast is it going after
  it goes up 2.0 m
• Use Work energy theorem DK WConservative
  WNon-conservative

3
Newtons Laws

• The table has a coefficient of kinetic friction
  of 0.350, the masses are m1 4.00 kg, m2
  1.00kg and m3 2.00kg.

m2
T1
m1
m3

• If m3 starts from rest how fast is it going after
  it goes up 2.0 m
• DK½ m1v1f2 ½ m2v2f2 ½ m3v3f2 -½ m1v1i2 -½
  m2v2i2 - ½ m3v3i2
• ½ (m1m2m3) v2
• WC. Wg(mass 1) Wg (mass 3) m1gh - m3gh
• WNC Wfriction Fx Dx - m2gm h
• ½ (m1m2m3) v2 m1gh - m3gh - m2gm h
• v2 2gh(m1-m3-m2m)/(m1m2m3)

4
Work, Energy Circular Motion

• A mass, 11 kg, slides down of a frictionless
  circular path of radius, 5.0 m, as shown in the
  figure. Initially it moves only vertically and,
  at the end, only horizontally (1/4 of a circle
  all told). Gravity, 10 m/s2, acts along the
  vertical. If the initial velocity is 2 m/s
  downward then(a) What is the work done by
  gravity on the mass? (b) What is the final
  speed of the mass when it reaches the bottom?
• (c) What is the normal force on the mass
• when it reaches the bottom (while still on
• the curved sections)?

5
Work, Energy Circular Motion

• A mass, 11 kg, slides down of a frictionless
  circular path of radius, 5.0 m, as shown in the
  figure. Initially it moves only vertically and,
  at the end, only horizontally (1/4 of a circle
  all told). Gravity, 10 m/s2, acts along the
  vertical. If the initial velocity is 2 m/s
  downward then(a) What is the work done by
  gravity on the mass? W mgR 11 x 10 x 5 550
  J
• (b) What is the final speed of the mass
• when it reaches the bottom (while still on the
  curve)?
• ½ mvf2 ½ m vi2 mgR 22 J 550 J 572 J
• vf (1144 / 11) ½ m/s

6
Work, Energy Circular Motion

• A mass, 11 kg, slides down of a frictionless
  circular path of radius, 5.0 m, as shown in the
  figure. Initially it moves only vertically and,
  at the end, only horizontally (1/4 of a circle
  all told). Gravity, 10 m/s2, acts along the
  vertical. If the initial velocity is 2 m/s
  downward then(c) What is the normal force on
  the mass
• when it reaches the bottom
• SFy m ac N mg m v2 /R
• N mg m v2 /R (110 11 x 1144/11) N
• 1254 N 1300 N

7
Exercise Work/Energy for Non-Conservative Forces

• An air track is at an angle of 30 with respect
  to horizontal. The cart (with mass 1.0 kg) is
  released 1.0 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)
• Notice the cart only bounces to a height of
  0.25 m

1. 2.5 J
2. 5.0 J
3. 10. J
4. -2.5 J
5. -5.0 J
6. -10. J

8
Exercise Work/Energy for Non-Conservative Forces

• Alternatively we could look at WnetDK
• Again Kfinal Kinitial 0
• WnetDK 0 Wgravity Wfriction
• (mg sin q ) (0.5 meter) Wfriction
• Wfriction -2.5 N m -2.5 J or (D)
• And the result is the same

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
9
Springs

• A Hookes Law spring with a spring constant of
  200 N/m is first stretched 3.0 m past its
  equilibrium distance and then is stretched 6.0 m
  more meters.
• How much work must be done to go from 3.0 m to
  9.0 m?
• W Ufinal-Uinitial ½ k (x-xeq)final2 -½ k
  (x-xeq)init2
• 100 (9)2 (3)2 J
  100(72) J 7200 J

10
Chapter 5 6 (Forces and Newtons Laws)
11
Chapter 6
12
Chapter 5 6
13
Chapter 5 6
Note Drag in air is proportional to v or v2 At
terminal velocity, drag force cancels out other
forces
14
Chapter 5 6
15
Chapter 7 8
16
Chapter 7 8
17
Chapter 7 8
Note Ethermal is the same as Einternal (Serway)
Note Wdissapative is the same as
Wnon-conservative
18
Chapter 7 8
s here refers to the path
19
Chapter 7 8
20
Work and Energy

• A block of mass m is connected by a spring to the
  ceiling. The block is held at a position where
  the spring is unstretched and then released. When
  released, the block
• (a) remains at rest.
• (b) oscillates about the unstretched position
• (c) oscillates about a position that is lower
  than the unstretched position
• (d) oscillates about a position that is higher
  than the unstretched position

21
Work and Energy

• A block of mass m is connected by a spring to the
  ceiling. The block is held at a position where
  the spring is unstretched and then released. When
  released, the block
• (a) remains at rest.
• (b) oscillates about the unstretched position
• (c) oscillates about a position that is lower
  than the unstretched position
• (d) oscillates about a position that is higher
  than the unstretched position

22
Work and Energy

• A mass is attached to a Hookes law spring on a
  horizontal surface as shown in the diagram below.
  When the spring is at its equilibrium length, the
  block is at position Y.
• When released from position X, how will the
  springs potential energy vary as the block moves
  from X to Y to Z ?
• (a) It will steadily increase from X to Z.
• (b) It will steadily decrease from X to Z.
• (c) It will increase from X to Y and decrease
  from Y to Z.
• (d) It will decrease from X to Y and increase
  from Y to Z.

23
Work and Energy

• A mass is attached to a Hookes law spring on a
  horizontal surface as shown in the diagram below.
  When the spring is at its equilibrium length, the
  block is at position Y.
• When released from position X, how will the
  springs potential energy vary as the block moves
  from X to Y to Z ?
• (a) It will steadily increase from X to Z.
• (b) It will steadily decrease from X to Z.
• (c) It will increase from X to Y and decrease
  from Y to Z.
• (d) It will decrease from X to Y and increase
  from Y to Z.

24
Work and Energy

• An object moves along a line under the influence
  of a single force. The area under the force vs.
  position graph represents
• (a) the impulse delivered to the object
• (b) the work done on the object.
• (c) the change in the velocity of the object.
• (d) the momentum of the object.

25
Work and Energy

• An object moves along a line under the influence
  of a single force. The area under the force vs.
  position graph represents
• (a) the impulse delivered to the object
• (b) the work done on the object.
• (c) the change in the velocity of the object.
• (d) the momentum of the object.

26
Concept problem

• At high speeds the drag force of the air is found
  to be proportional to the square of a cars
  speed. Assume that at 60 mph that 100 of cars
  power is being used against wind resistance
  (i.e., there are no other non-conservative
  forces.) In terms of the ratio P(120 mph) / P
  (60 mph), how much more power will the cars
  engine need to provide if this car is to travel
  at 120 mph?
• 2
• 23/2
• 4
• 25/2
• 8
• P F v v3 so ratio is 8

27
Work and Energy

• A block slides along a frictionless surface
  before colliding with a spring. The block is
  brought momentarily to rest by the spring after
  traveling some distance. The four scenarios shown
  in the diagrams below are labeled with the mass
  of the block, the initial speed of the block, and
  the spring constant.
• Rank the scenarios in order of the distance the
  block travels, listing the largest distance
  first.
• (a) B , A , C D
• (b) B , C , A , D
• (c) B , C D , A
• (d) C B, A , D
• (e) C B D , A

28
Work and Energy

• A block slides along a frictionless surface
  before colliding with a spring. The block is
  brought momentarily to rest by the spring after
  traveling some distance. The four scenarios shown
  in the diagrams below are labeled with the mass
  of the block, the initial speed of the block, and
  the spring constant.
• Rank the scenarios in order of the distance the
  block travels, listing the largest distance
  first.
• (a) B , A , C D
• (b) B , C , A , D
• (c) B , C D , A
• (d) C B, A , D
• (e) C B D , A

29
Work and Forces

• A 25.0 kg chair is pushed 2.00 m at constant
  speed along a horizontal surface with a constant
  force acting at 30.0 degrees below the
  horizontal. If the friction force between the
  chair and the surface is 55.4 N, what is the work
  done by the pushing force?
• (a) 85 J
• (b) 98 J
• (c) 111 J
• (d) 113 J
• (e) 128 J

30
Work and Forces

• A 25.0 kg chair is pushed 2.00 m at constant
  speed along a horizontal surface with a constant
  force acting at 30.0 degrees below the
  horizontal. If the friction force between the
  chair and the surface is 55.4 N, what is the work
  done by the pushing force?
• (a) 85 J
• (b) 98 J
• (c) 111 J
• (d) 113 J
• (e) 128 J

31
Work and Power

• A 100 kg elevator is carrying 6 people, each
  weighing 70 kg. They all want to travel to the
  top floor, 75 m from the floor they entered at.
  How much power will the elevator motor supply to
  lift this in 45 seconds at constant speed?
• (a) 1.2 102 W
• (b) 7.0 102 W
• (c) 8.7 102 W
• (d) 6.9 103 W
• (e) 8.5 103 W

32
Work and Power

• A 100 kg elevator is carrying 6 people, each
  weighing 70 kg. They all want to travel to the
  top floor, 75 m from the floor they entered at.
  How much power will the elevator motor supply to
  lift this in 45 seconds at constant speed?
• (a) 1.2 102 W
• (b) 7.0 102 W
• (c) 8.7 102 W
• (d) 6.9 103 W
• (e) 8.5 103 W

33
Newtons Laws

• Two sleds are hooked together in tandem. The
  front sled is twice as massive as the rear sled.
  The sleds are pulled along a frictionless surface
  by a force F, applied to the more massive sled.
  The tension in the rope between the sleds is T.
  Determine the ratio of the magnitudes of the two
  forces, T/F.
• (a) 0.33
• (b) 0.50
• (c) 0.67
• (d) 1.5
• (e) 2.0
• (f) 3.0

34
Newtons Laws

• A factory worker raises a 100. kg crate at a
  constant rate using a frictionless pulley system,
  as shown in the diagram. The mass of the pulleys
  and rope are negligible. (assume g 9.8 m/s2)
• With what force is the worker pulling down on the
  rope?
• (a) 245 N
• (b) 327 N
• (c) 490 N
• (d) 980 N
• (e) 1960 N

35
Work and Energy

• A 6.0 kg block is pushed up against an ideal
  Hookes law spring (of spring constant 3750 N/m )
  until the spring is compressed a distance x. When
  it is released, the block travels along a track
  from one level to a higher level, by moving
  through an intermediate valley (as shown in the
  diagram). The track is frictionless until the
  block reaches the higher level. There is a
  frictional force stops the block in a distance of
  1.2 m. If the coefficient of friction between the
  block and the surface is 0.60, what is x ? (Let g
  9.81 m/s2 )
• (a) 0.11 m
• (b) 0.24 m
• (c) 0.39 m
• (d) 0.48 m
• (e) 0.56 m

36
Work and Energy

• A 6.0 kg block is pushed up against an ideal
  Hookes law spring (of spring constant 3750 N/m )
  until the spring is compressed a distance x. When
  it is released, the block travels along a track
  from one level to a higher level, by moving
  through an intermediate valley (as shown in the
  diagram). The track is frictionless until the
  block reaches the higher level. There is a
  frictional force stops the block in a distance of
  1.2 m. If the coefficient of friction between the
  block and the surface is 0.60, what is x ? (Let g
  9.81 m/s2 )
• (a) 0.11 m
• (b) 0.24 m
• (c) 0.39 m
• (d) 0.48 m
• (e) 0.56 m

37
Conceptual Problem
A bird sits in a birdfeeder suspended from a tree
by a wire, as shown in the diagram at left.
Let WB and WF be the weight of the bird and the
feeder respectively. Let T be the tension in the
wire and N be the normal force of the feeder on
the bird. Which of the following free-body
diagrams best represents the birdfeeder? (The
force vectors are not drawn to scale and are only
meant to show the direction, not the magnitude,
of each force.)
38
Conceptual Problem
A block is pushed up a 20º ramp by a 15 N force
which may be applied either horizontally (P1) or
parallel to the ramp (P2). How does the
magnitude of the normal force N depend on the
direction of P?

• (A) N will be smaller if P is horizontal than
  if it is parallel the ramp.
• (B) N will be larger if P is horizontal than if
  it is parallel to the ramp.
• (C) N will be the same in both cases.
• (D) The answer will depend on the coefficient of
  friction.

20
39
Conceptual Problem
A cart on a roller-coaster rolls down the track
shown below. As the cart rolls beyond the point
shown, what happens to its speed and acceleration
in the direction of motion?
A. Both decrease. B. The speed decreases, but the
acceleration increases. C. Both remain
constant. D. The speed increases, but
acceleration decreases. E. Both increase. F. Other
40
Sample Problem

• A 200 kg wood crate sits in the back of a truck.
  The coefficients of friction between the crate
  and the truck are µs 0.9 and µk 0.5.
• The truck starts moving up a 20 slope. What
  is the maximum acceleration the truck can have
  without the crate slipping out the back?
• Solving
• Visualize the problem, Draw a picture if
  necessary
• Identify the system and make a Free Body Diagram
• Choose an appropriate coordinate system
• Apply Newtons Laws with conditional constraints
  (friction)
• Solve

41
Sample Problem

• You have been hired to measure the coefficients
  of friction for the newly discovered substance
  jelloium. Today you will measure the coefficient
  of kinetic friction for jelloium sliding on
  steel. To do so, you pull a 200 g chunk of
  jelloium across a horizontal steel table with a
  constant string tension of 1.00 N. A motion
  detector records the motion and displays the
  graph shown.
• What is the value of µk for jelloium on steel?

42
Sample Problem

• S Fx ma F - ff F - mk N F - mk mg
• S Fy 0 N mg
• mk (F - ma) / mg x ½ a t2 ? 0.80 m
  ½ a 4 s2
• a 0.40 m/s2
• mk (1.00 - 0.20 0.40 ) / (0.20 10.) 0.46

43
Exercise Newtons 2nd Law
A force of 2 Newtons acts on a cart that is
initially at rest on an air track with no air and
pushed for 1 second. Because there is friction
(no air), the cart stops immediately after I
finish pushing. It has traveled a distance, D.
Next, the force of 2 Newtons acts again but is
applied for 2 seconds. The new distance the
cart moves relative to D is

1. 8 x as far
2. 4 x as far
3. 2 x as far
4. 1/4 x as far

44
Exercise Solution
We know that under constant acceleration, Dx
a (Dt)2 /2 (when v00)
Here Dt22Dt1, F2 F1 ? a2 a1
(B) 4 x as long
45
Sample exam problem

• A small block moves along a frictionless incline
  which is 45 from horizontal. Gravity acts down
  at 10 m/s2. There is a massless cord pulling on
  the block. The cord runs parallel to the incline
  over a pulley and then straight down. There is
  tension, T1, in the cord which accelerates the
  block at 2.0 m/s2 up the incline. The pulley is
  suspended with a second cord with tension, T2.

A. What is the tension magnitude, T1, in the 1st
cord? B. What is the tension magnitude,T2, in the
2nd cord?
46
Sample exam problem

• a 2.0 m/s2 up the incline.

What is the tension magnitude, T1, in the 1st
cord? Use a FBD! Along the block surface S Fx
m ax -mg sin q T T 5 x 2 N 5 x 10 x
0.7071 N (10 35) N 45 N
47
Sample exam problem

• a 0.0 m/s2 at the pulley.

What is the tension magnitude,T2, in the 2nd
cord? Use a FBD!
48
Sample exam problem

• You have a 2.0 kg block that moves on a linear
  path on a horizontal surface. The coefficient of
  kinetic friction between the block and the path
  is µk. Attached to the block is a horizontally
  mounted massless string as shown in the figure
  below. The block includes an accelerometer which
  records acceleration vs. time. As you increase
  the tension in the rope the block experiences an
  increasingly positive acceleration. At some point
  in time the rope snaps and then the block slides
  to a stop (at a time of 10 seconds). Gravity,
  with g 10 m/s2, acts downward.

49
Sample exam problem

• A. At what time does the string break and, in one
  sentence, explain your reasoning?
• B. What speed did the block have when the string
  broke?
• C. What is the value of µk?
• D. Using µk above (or a value of 0.25 if you
  dont have one), what was the tension in the
  string at t 2 seconds?

50
Sample exam problem

• B. What speed did the block have when the string
  broke?
• Dont know initial v (t0) so cant integrate
  area at t lt 4 sec.
• vf 0 m/s and from t 4 to 10 sec (6 second) a
  - 2 m/s2
• 0 vi a t vi 2 x 6 m/s ? vi 12 m/s

51
Sample exam problem

• C. What is the value of µk? Use a FBD!
• S Fx m ax - fk - µk N
• S Fy 0 mg N ? N mg
• So m ax - fk - µk mg ? µk - ax / g -
  (-2)/10 0.20

52
Sample exam problem

• D. What was the tension in the string at t 2
  seconds?
• Again a FBD!
• S Fx m ax - fk T
• S Fy 0 mg N ? N mg
• T fk m ax (0.20 x 2 x 10 2 x 3 ) N
  10 N

53
Sample exam problem

• A 5.0 kg block with an equilateral triangular
  cross-section lies as shown in a 60 frictionless
  groove. Gravity, with g 10. m/s2, acts
  downward.
• You may use the following relationships
• cos 60 sin 30 0.50
• cos 30 sin 60 0.87.
• A. Draw a free body diagram showing all the
  forces acting on the block.
• B. What are the magnitudes of the normal forces
  on the block associated with the two sides of
  contact (A to B and B to C)?

N
N
q
q
mg
Use a FBD! ( q 30) S Fx 0 -N cos 30
N cos 30 ? N N S Fy 0 -mg N sin
30 N sin 30 ? N 50 N This is why wedges
are good for splitting things