1. Review: using F=ma Force in circular motion
𝑎 𝑐 and 𝑎 𝑡 in curvilinear motion (e.g. non-uniform circular motion)
Note on other important forces
Propulsion
Resistive force (Drag in air/liquid), terminal velocicty 𝑣 𝑇
Conservative force, spring force Potential energy

2. Relevant concepts:
“inertia” and force
a) Inertial reference frame
b) accelerated frame of the car
- fictitious force

3. Rolling friction

5. Conical Pendulum (e.g. ชิงช้าหมุน)
The object is in equilibrium in the vertical direction.
It undergoes uniform circular motion in the horizontal direction.
∑𝐹𝑦 = 0 → 𝑇 cos 𝜃 = 𝑚𝑔
∑𝐹𝑥 = 𝑇 sin⁡𝜃 = 𝑚 𝑎𝑐
v is independent of m

6. Car traveling in a banked curve
Design the curve with no friction
in equilibrium in the vertical direction.
in uniform circular motion in the horizontal direction a component of the normal force supplies the centripetal force.
The angle of bank is
Note:
The banking angle is independent of the mass of the vehicle.
If the car rounds the curve at less than the design speed, friction is necessary to keep it from sliding down the bank.
If the car rounds the curve at more than the design speed, friction is necessary to keep it from sliding up the bank.

7. Car traveling in a horizontal (Flat) Curve
uniform circular motion in the horizontal direction.
in equilibrium in the vertical direction.
The force of static friction supplies the centripetal force.
The maximum speed at which the car can negotiate the curve is:
Note: this does not depend on the mass of the car.
Section 6.1

8. Ferris Wheel 𝑭 𝒏𝒆𝒕 𝑭 𝒏𝒆𝒕 = 𝑚 𝑣 2 𝑅 (− 𝒓 ) 𝑭 𝒏𝒆𝒕
Uniform circular motion with constant speed v (controlled by the motor)
Under gravity, the child feels apparent weight differently at top and bottom
𝑭 𝒏𝒆𝒕
𝑭 𝒏𝒆𝒕 = 𝑚 𝑣 2 𝑅 (− 𝒓 )
𝑭 𝒏𝒆𝒕
Section 6.1

9. Ferris Wheel (2)
At the bottom of the loop, the upward force (the normal) experienced by the object is greater than its weight.
At the top of the circle, the force exerted on the object is less than its weight.
Section 6.1

10. Non-uniform Circular Motion
If the speed also changes in magnitude
there is non-zero tangential acceleration
Section 6.2

11. Vertical Circle with Non-Uniform Speed
Launch by shooting (say, from the bottom at initial velocity 𝒗 𝒃𝒐𝒕 ). The string restricts motion to a circle of radius R (1 DOF)
What is T and acceleration?
Tension depends on position 𝜃(𝑡) and speed 𝑣 𝑡 = d𝜃 𝑑𝑡 𝑅 ,both varying with time
The tension at the bottom is a maximum.
The tension at the top is a minimum.
Requires critical speed 𝑣 𝑡𝑜𝑝 ≥ 𝑅𝑔 to complete the circle ( 𝑇 𝑡𝑜𝑝 ≥ 0 at 𝜃 =𝜋)
Section 6.2

12. Resistive force (or drag force in air/fluid) and terminal velocity 𝑣 𝑇 (this topic is not in the test)

13. 1. Linear (Stokes law in liquid) 𝑭 𝑫 ∝− 𝒗 𝐹 𝐷 = −𝑏𝑣
Resistive force 𝑭 𝑫 could be of either form(linear or quadratic to 𝑣), but always opposite direction to motion (− 𝒗 )
1. Linear (Stokes law in liquid)
𝑭 𝑫 ∝− 𝒗
𝐹 𝐷 = −𝑏𝑣
2. Quadratic (air resistance)
𝑭 𝑫 ∝− 𝑣 2
𝐹 𝐷 = − 1 2 𝐶 𝐷 𝜌𝐴 𝑣 2
𝐶 𝐷 is the drag coefficient (empirical, no dimension, typically 𝐶 𝐷 ≈ 1 2 ).
𝜌 is the density of air.
𝐴 is the cross-sectional area of the object
𝑣 𝑇 ≈ 4𝑚𝑔 𝜌𝐴

15. Lecture 4
Work and Energy

16. Outline Work done by a constant force
Projection and scalar product of vectors
Force that results in positive work.
negative work? forces that do no work?
Work done by varying force with displacement
Work-energy theorem

17. 6.1 Work Done by a Constant Force

18. 6.1 Work Done by a Constant Force

19. Work, cont.
W = F Dr cos q
A force does no work on the object if the force does not move through a displacement.
The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application.
Section 7.2

20. 6.1 Work Done by a Constant Force

21. 6.1 Work Done by a Constant Force
Example 1 Pulling a Suitcase-on-Wheels
Find the work done if the force is 45.0-N, the angle is 50.0
degrees, and the displacement is 75.0 m.

22. 6.1 Work Done by a Constant Force

23. 6.1 Work Done by a Constant Force
Example 3 Accelerating a Crate
The truck is accelerating at
a rate of m/s2. The mass
of the crate is 120-kg and it
does not slip. The magnitude of
the displacement is 65 m.
What is the total work done on
the crate by all of the forces
acting on it?

24. 6.1 Work Done by a Constant Force
The angle between the displacement
and the friction force is 0 degrees.

25. Scalar Product of Two Vectors
Section 7.3

26. Scalar Product, cont The scalar product is commutative.
The scalar product obeys the distributive law of multiplication.
Section 7.3

27. Dot Products of Unit Vectors
Using component form with vectors: In the special case where
Section 7.3

28. 6.2 The Work-Energy Theorem and Kinetic Energy
Consider a constant net external force acting on an object.
The object is displaced a distance s, in the same direction as
the net force.
The work is simply

29. 6.2 The Work-Energy Theorem and Kinetic Energy
DEFINITION OF KINETIC ENERGY
The kinetic energy KE of and object with mass m
and speed v is given by

30. 6.2 The Work-Energy Theorem and Kinetic Energy
When a net external force does work on and object, the kinetic
energy of the object changes according to

31. 6.2 The Work-Energy Theorem and Kinetic Energy
Example 4 Deep Space 1
The mass of the space probe is 474-kg and its initial velocity
is 275 m/s. If the 56.0-mN force acts on the probe through a
displacement of 2.42×109m, what is its final speed?

32. 6.2 The Work-Energy Theorem and Kinetic Energy

33. 6.2 The Work-Energy Theorem and Kinetic Energy

34. 6.2 The Work-Energy Theorem and Kinetic Energy
In this case the net force is

35. 6.2 The Work-Energy Theorem and Kinetic Energy
Conceptual Example 6 Work and Kinetic Energy
A satellite is moving about the earth
in a circular orbit and an elliptical orbit.
For these two orbits, determine whether
the kinetic energy of the satellite
changes during the motion.
W<0
W>0

36. 6.3 Gravitational Potential Energy

37. 6.3 Gravitational Potential Energy
Work due to gravity is independent of path!
We call this phenomenon “conservative force”
 We can define “potential energy” based on this force

38. 6.3 Gravitational Potential Energy
Example 7 A Gymnast on a Trampoline
The gymnast leaves the trampoline at an initial height of 1.20 m
and reaches a maximum height of 4.80 m before falling back
down. What was the initial speed of the gymnast?

39. 6.3 Gravitational Potential Energy
DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY
The gravitational potential energy PE is the energy that an
object of mass m has by virtue of its position relative to the
surface of the earth. That position is measured by the height
h of the object relative to an arbitrary zero level:

40. 6.4 Conservative Versus Nonconservative Forces
DEFINITION OF A CONSERVATIVE FORCE
Version 1 A force is conservative when the work it does
on a moving object is independent of the path between the
object’s initial and final positions.
Version 2 A force is conservative when it does no work
on an object moving around a closed path, starting and
finishing at the same point.

41. 6.4 Conservative Versus Nonconservative Forces

42. 6.4 Conservative Versus Nonconservative Forces
Version 1 A force is conservative when the work it does
on a moving object is independent of the path between the
object’s initial and final positions.

43. 6.4 Conservative Versus Nonconservative Forces
Version 2 A force is conservative when it does no work
on an object moving around a closed path, starting and
finishing at the same point.

45. 6.4 Conservative Versus Nonconservative Forces
An example of a nonconservative force is the kinetic
frictional force.
The work done by the kinetic frictional force is always negative.
Thus, it is impossible for the work it does on an object that
moves around a closed path to be zero.
The concept of potential energy is not defined for a
nonconservative force.

46. 6.4 Conservative Versus Nonconservative Forces
In normal situations both conservative and nonconservative
forces act simultaneously on an object, so the work done by
the net external force can be written as

47. Spring force and Spring (Elastic) Potential Energy

48. 6.4 Conservative Versus Nonconservative Forces
Work done by conservative force
Work done by non-conservative force
THE WORK-ENERGY THEOREM

49. 6.5 The Conservation of Mechanical Energy
If the net work on an object by nonconservative forces
is zero, then its energy does not change:

50. 6.5 The Conservation of Mechanical Energy
THE PRINCIPLE OF CONSERVATION OF
MECHANICAL ENERGY
The total mechanical energy (E = KE + PE) of an object
remains constant as the object moves, provided that the net
work done by external nononservative forces is zero.

51. 6.5 The Conservation of Mechanical Energy

52. 6.5 The Conservation of Mechanical Energy
Example 8 A Daredevil Motorcyclist
A motorcyclist is trying to leap across the canyon by driving
horizontally off a cliff 38.0 m/s. Ignoring air resistance, find
the speed with which the cycle strikes the ground on the other
side.

53. 6.5 The Conservation of Mechanical Energy

54. 6.5 The Conservation of Mechanical Energy

55. 6.5 The Conservation of Mechanical Energy
Conceptual Example 9 The Favorite Swimming Hole
The person starts from rest, with the rope
held in the horizontal position,
swings downward, and then lets
go of the rope. Three forces
act on him: his weight, the
tension in the rope, and the
force of air resistance.
Can the principle of
conservation of energy
be used to calculate his
final speed?

56. DEFINITION OF AVERAGE POWER
Average power is the rate at which work is done, and it
is obtained by dividing the work by the time required to
perform the work.

57. 6.7 Power

58. 6.7 Power

59. 6.8 Other Forms of Energy and the Conservation of Energy
THE PRINCIPLE OF CONSERVATION OF ENERGY
Energy can neither be created not destroyed, but can
only be converted from one form to another.

60. Thermal energy (heat) as internal energy of materials
Where does work due to nonconservative force (Wnc) go to, other than kinetic energy?
Thermal energy (heat) as internal energy of materials
This means energy of the universe (isolated system) is always conserved.
When we say “not energy conserving” we mean just in the sub-system.

62. Lecture 5
Impulse and Momentum

63. 7.1 The Impulse-Momentum Theorem
There are many situations when the
force on an object is not constant.

64. 7.1 The Impulse-Momentum Theorem
DEFINITION OF IMPULSE
The impulse of a force is the product of the average
force and the time interval during which the force acts:
Impulse is a vector quantity and has the same direction
as the average force.

65. 7.1 The Impulse-Momentum Theorem

68. 7.1 The Impulse-Momentum Theorem
DEFINITION OF LINEAR MOMENTUM
The linear momentum of an object is the product
of the object’s mass times its velocity:
Linear momentum is a vector quantity and has the same
direction as the velocity.

69. 7.1 The Impulse-Momentum Theorem

70. 7.1 The Impulse-Momentum Theorem
When a net force acts on an object, the impulse of
this force is equal to the change in the momentum
of the object
impulse
final momentum
initial momentum

71. 7.1 The Impulse-Momentum Theorem
Example 2 A Rain Storm
Rain comes down with a velocity of -15 m/s and hits the
roof of a car. The mass of rain per second that strikes
the roof of the car is kg/s. Assuming that rain comes
to rest upon striking the car, find the average force
exerted by the rain on the roof.

72. 7.1 The Impulse-Momentum Theorem
Neglecting the weight of
the raindrops, the net force
on a raindrop is simply the
force on the raindrop due to
the roof.

73. 7.1 The Impulse-Momentum Theorem
Conceptual Example 3 Hailstones Versus Raindrops
Instead of rain, suppose hail is falling. Unlike rain, hail usually
bounces off the roof of the car.
If hail fell instead of rain, would the force be smaller than,
equal to, or greater than that calculated in Example 2?

75. 7.2 The Principle of Conservation of Linear Momentum
If the sum of the external forces is zero, then
PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM
The total linear momentum of an isolated system is constant
(conserved). An isolated system is one for which the sum of
the average external forces acting on the system is zero.

76. 7.2 The Principle of Conservation of Linear Momentum
Conceptual Example 4 Is the Total Momentum Conserved?
Imagine two balls colliding on a billiard
table that is friction-free. Use the momentum
conservation principle in answering the
following questions. (a) Is the total momentum
of the two-ball system the same before
and after the collision? (b) Answer
part (a) for a system that contains only
one of the two colliding
balls.

77. 7.2 The Principle of Conservation of Linear Momentum
Example 6 Ice Skaters
Starting from rest, two skaters
push off against each other on
ice where friction is negligible.
One is a 54-kg woman and
one is a 88-kg man. The woman
moves away with a speed of
+2.5 m/s. Find the recoil velocity
of the man.

78. 7.2 The Principle of Conservation of Linear Momentum

79. 7.2 The Principle of Conservation of Linear Momentum
Applying the Principle of Conservation of Linear Momentum
1. Decide which objects are included in the system.
2. Relative to the system, identify the internal and external forces.
3. Verify that the system is isolated.
4. Set the final momentum of the system equal to its initial momentum.
Remember that momentum is a vector.

80. การชนกัน (collision)
- ในการชนส่วนใหญ่ เราไม่ทราบแรงชน จึงไม่สามารถวิเคราะห์ การเคลื่อนที่โดยใช้กฎของนิวตันได้ - แต่หากทราบลักษณะการเคลื่อนที่ (ความเร็ว) การวิเคราะห์การเคลื่อนที่ สำหรับการชนสามารถใช้หลักอนุรักษ์โมเมนตัมได้ - กรณีที่เป็นการชนแบบยืดหยุ่น เป็นการชนแบบไม่สูญเสียพลังงานจลน์ สามารถใช้หลักอนุรักษ์พลังงานจลน์ซึ่ง หมายถึง พลังงานจลน์รวมของระบบก่อนและหลังชนมีค่าเท่ากัน - กรณีที่เป็นการชนแบบไม่ยืดหยุ่น พลังงานจลน์ของระบบจะลดลง เนื่องจากพลังงานจลน์เปลี่ยนรูปเป็น เสียง ความร้อน เป็นต้น เช่น หลังชนวัตถุติดกัน

81. 7.3 Collisions in One Dimension
The total linear momentum is conserved when two objects
collide, provided they constitute an isolated system.
Elastic collision -- One in which the total kinetic
energy of the system after the collision is equal to
the total kinetic energy before the collision.
Inelastic collision -- One in which the total kinetic
energy of the system after the collision is not equal
to the total kinetic energy before the collision; if the
objects stick together after colliding, the collision is
said to be completely inelastic.

82. Example of elastic collision in 1D

83. 7.3 Collisions in One Dimension
Example 8 A Ballistic Pendulum
The mass of the block of wood
is 2.50-kg and the mass of the
bullet is kg. The block
swings to a maximum height of
0.650 m above the initial position.
Find the initial speed of the
bullet.

84. 7.3 Collisions in One Dimension
Apply conservation of momentum
to the collision:
Inelastic collision

85. 7.3 Collisions in One Dimension
Applying conservation of energy
to the swinging motion:

86. 7.3 Collisions in One Dimension

87. The center of mass is a point that represents the average location for
the total mass of a system.

88. 7.5 Center of Mass

89. In an isolated system, the total linear momentum does not change,
7.5 Center of Mass
In an isolated system, the total linear momentum does not change,
therefore the velocity of the center of mass does not change.

90. 7.5 Center of Mass
BEFORE
AFTER