1. Simple Harmonic Motion and Elasticity

2. Force is proportional to displacement
For small displacements, the force required to stretch or compress a spring is directly proportional to the displacement x, or The constant k is called the spring constant or stiffness of the spring. A spring behaves according to the above equation is said to be an ideal spring.

3. 10.1 The Ideal Spring and Simple Harmonic Motion
spring constant
Units: N/m

4. 10.1 The Ideal Spring and Simple Harmonic Motion
Example-1: A Tire Pressure Gauge
The spring constant of the spring
is 320 N/m and the bar indicator
extends 2.0 cm.
What force does the
air in the tire apply to the spring?

5. 10.1 The Ideal Spring and Simple Harmonic Motion

6. Tire Pressure?
Tire pressure varies from 2 bar to 2.2 bar.
Or, 28 psi to 32 psi
1 bar = 100 kPa = Pa = N/m2
Pressure = Force per unit Area
P = F/A
What is the area of the Tire Pressure Gauge?

7. 10.1 The Ideal Spring and Simple Harmonic Motion
Conceptual Example-2: Are Shorter Springs Stiffer?
A 10-coil spring has a spring constant k. If the spring is
cut in half, so there are two 5-coil springs, what is the spring
constant of each of the smaller springs?

8. Shorter Springs are Stiffer
The spring constant of each 5-coil spring is 2k. Spring constant  1/# of coils k for 10 coils 2k for 5 coils 10k for 1 coil

9. Restoring Force
To stretch or compress a spring, a force must be applied to it. In accord with Newton’s third law, the spring exert an oppositely directed force of equal magnitude. This reaction force is applied by the spring to the agent that does the pulling or pushing. The reaction force is also called a “restoring force”. Fx = - kx The reaction force always points in a direction opposite to the displacement of the spring from its unstrained length.

10. Restoring Force
The restoring force of a spring can also contribute to the net external force.

11. 10.1 The Ideal Spring and Simple Harmonic Motion
HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING
The restoring force on an ideal spring is

12. Why it is called Restoring Force
An object of mass m is attached to a spring on a frictionless table. In part A, the spring has been stretched to the right, so the spring exerts the left-ward pointing force Fx. When the object is released, this force pulls it to the left, restoring it toward its equilibrium position.

13. Restoring Force causes Simple Harmonic Motion
When the object is released, the restoring force pulls it to its equilibrium position. In accord with Newton’s first law, the moving object has inertia and coasts beyond the equilibrium position, compressing the spring as in part B. The restoring force exerted by the spring now points to the right and after bringing the object to a momentary halt, acts to restore the object to its equilibrium position. Since no friction acts on the object the back-and-forth motion repeats itself.
When the restoring force has the mathematical form given by Fx = -kx, the type of friction-free motion is designated as “simple harmonic motion”.

14. Simple Harmonic Motion
By attaching a pen to the object and moving a strip of paper past it at a steady rate, we can record the position of the vibrating object as time passes. The shape of this graph is characteristic of simple harmonic motion and is called sinusoidal.

15. Simple Harmonic Motion
Simple harmonic motion, like any motion, can be described in terms of
displacement,
velocity, and
acceleration.

16. 10.2 Simple Harmonic Motion and the Reference Circle
DISPLACEMENT

17. 10.2 Simple Harmonic Motion and the Reference Circle
Radius = A
The displacement of the shadow, x is just the projection of the radius A onto the x-axis:
Where  = t
, the angular speed in rad/s

18. Angular Speed,   = /t rad/s = t rad
Angular displacement () for one cycle is 2 rad in T.
So,  = 2/T
 = 2
Because  is directly proportional to the frequency ,  is often called the angular frequency.

19. 10.2 Simple Harmonic Motion and the Reference Circle
amplitude A: the maximum displacement
period T: the time required to complete one cycle
frequency f: the number of cycles per second (measured in Hz)

20. 10.2 Simple Harmonic Motion and the Reference Circle
VELOCITY
Tangential velocity, VT
= Radius x angular velocity
Velocity of the shadow, Vx
= X component of VT
Maximum Velocity
of the shadow, Vx = A

21. 10.2 Simple Harmonic Motion and the Reference Circle
Example 3 The Maximum Speed of a Loudspeaker Diaphragm
The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm.
What is the maximum speed of the diaphragm?
Where in the motion does this maximum speed occur?

22. 10.2 Simple Harmonic Motion and the Reference Circle
The maximum speed
occurs midway between
the ends of its motion.

23. Simple harmonic motion?
Is the motion of the lighted bulb simple harmonic motion, when each lights for 0.5s in sequence?

24. 10.2 Simple Harmonic Motion and the Reference Circle
ACCELERATION
The ball on the reference circle moves in uniform circular motion, and, therefore, has centripetal acceleration ac that points toward the centre of the circle.
The acceleration ax of the shadow is the x component of the centripetal acceleration ac .

25. Maximum Acceleration
A loudspeaker vibrating at 1kHz with an amplitude of 0.20mm has a maximum acceleration of
amax = A2 = 7.9 x 103 m/s2
Maximum acceleration occurs when the force acting on the diaphragm is a maximum.
The maximum force arises when the diaphragm is at the ends of its path.

26. Frequency of Vibration
With the aid of the Newton’s second law , it is possible to determine the frequency at which an object of mass m vibrates on a spring.
Mass of the spring is negligible
Only force acting is the restoring force.

27. 10.2 Simple Harmonic Motion and the Reference Circle
FREQUENCY OF VIBRATION
Larger spring constants k
and smaller masses m
result in larger frequencies.

28. 10.2 Simple Harmonic Motion and the Reference Circle
Example 6 A Body Mass Measurement Device
The device consists of a spring-mounted chair in which the astronaut
sits. The spring has a spring constant of 606 N/m and the mass of
the chair is 12.0 kg. The measured
period is 2.41 s. Find the mass of the
astronaut.

29. 10.2 Simple Harmonic Motion and the Reference Circle

30. 10.3 Energy and Simple Harmonic Motion
A compressed spring can do work.

31. Average magnitude of Force
Spring force at x0 is kx0
Spring force at xf is kxf
Average Fx = ½(kx0+kxf)

32. 10.3 Energy and Simple Harmonic Motion
Work done by the average spring force:
Final elastic PE
Initial elastic PE

33. 10.3 Energy and Simple Harmonic Motion
DEFINITION OF ELASTIC POTENTIAL ENERGY
The elastic potential energy is the energy that a spring
has by virtue of being stretched or compressed. For an
ideal spring, the elastic potential energy is
SI Unit of Elastic Potential Energy: joule (J)

34. 10.3 Energy and Simple Harmonic Motion
Conceptual Example 8 Changing the Mass of a Simple
Harmonic Oscilator
The box rests on a horizontal, frictionless
surface. The spring is stretched to x=A
and released. When the box is passing
through x=0, a second box of the same
mass is attached to it. Discuss what
happens to the (a) maximum speed
(b) amplitude (c) angular frequency.

35. Doubling the Mass of a Simple Harmonic Oscillator
At x=0m, a second box of the same mass and speed vmax is attached.
So, the max KE is doubled as mass is doubled.
So, when the spring compresses it will have double the PEe .
As PEe is doubled max amplitude will be 2 times

36. Doubling the Mass causes max amplitude to 2 times
Before adding the second mass, the displacement x1 is related to the PE as:
As PEe is doubled the new amplitude is x2

37. Doubling the Mass of a Simple Harmonic Oscillator
Angular frequency is
So, a doubling of mass will cause the angular frequency reduce to
(1/ 2 ) 

38. 10.3 Energy and Simple Harmonic Motion
Example 9 A falling ball on a vertical Spring
A 0.20-kg ball is attached to a vertical spring. The spring constant
is 28 N/m. When released from rest, how far does the ball fall
before being brought to a momentary stop by the spring?

39. 10.3 Energy and Simple Harmonic Motion

40. A simple pendulum consists of
10.4 The Pendulum
A simple pendulum consists of
a particle mass m attached to a frictionless
pivot by a cable of negligible mass.

41. simple pendulum
The force of gravity is responsible for the back-and-forth rotation about the axis at P.
A net torque is required to change the angular speed. The gravitational force mg produces the torque.
= - (mg)l - sign to represent the restoring force
= - (mg)L
= - k’  where k’ = mgL
Same as F = - kx
So,  = (k’/m) = (mgL/m) = (mgL/I) in rotational motion in place of mass moment of inertia ‘I’ will appear

42. simple pendulum
So,  = (k’/m) = (mgL/m) = (mgL/I) in rotational motion in place of mass moment of inertia ‘I’ will appear
The moment of inertia of a particle of mass m, rotating at a radius L about an axis is
I = mL2
Frequency of a simple pendulum is
Which does not depend on the mass of the particle.

43. Determine the length of a simple pendulum that will
10.4 The Pendulum
Example 10 Keeping Time
Determine the length of a simple pendulum that will
swing back and forth in simple harmonic motion with
a period of 1.00 s.

44. 10.5 Damped Harmonic Motion
In simple harmonic motion, an object oscillated
with a constant amplitude.
In reality, friction or some other energy
dissipating mechanism is always present
and the amplitude decreases as time
passes.
This is referred to as damped harmonic
motion.

45. 10.5 Damped Harmonic Motion
simple harmonic motion
2 & 3) underdamped
critically damped
5) overdamped

46. 10.6 Driven Harmonic Motion and Resonance
When a force is applied to an oscillating system at all times,
the result is driven harmonic motion.
Here, the driving force has the same frequency as the
spring system and always points in the direction of the
object’s velocity.

47. 10.6 Driven Harmonic Motion and Resonance
Resonance is the condition in which a time-dependent force can transmit
large amounts of energy to an oscillating object, leading to a large amplitude
motion.
Resonance occurs when the frequency of the force matches a natural
frequency at which the object will oscillate.

48. Elastic Deformation
All materials become distorted when they are squeezed or stretched.
Those materials return to their original shape when the deforming force is removed, such materials are called “elastic”.

49. Atomic View of Elastic materials
10.7 Elastic Deformation
Atomic View of Elastic materials
Because of these atomic-level “springs”, a material tends to
return to its initial shape once forces have been removed.
ATOMS
FORCES

50. Stretching force
Magnitude of the deforming force can be expressed as follows, provided the amount of stretch or compression is small compared to the original length of the object.

51. STRETCHING, COMPRESSION, AND YOUNG’S MODULUS
10.7 Elastic Deformation
STRETCHING, COMPRESSION, AND YOUNG’S MODULUS
Y is the proportionality constant called Young’s modulus
Young’s modulus has the units of pressure: N/m2

52. Young’s modulus
The magnitude of the force is proportional to the fractional increase (or decrease) in length L/L0, rather than the absolute change L.
Young’s modulus depends on the nature of the material.
For a given force, the material with higher Y undergoes the smaller change in length.
This force also called the “tensile” force,
because they cause a tension in the material

53. 10.7 Elastic Deformation

54. Example 12 Bone Compression
10.7 Elastic Deformation
Example 12 Bone Compression
In a circus act, a performer supports the combined weight (1080 N) of
a number of colleagues. Each thighbone of this performer has a length
of 0.55 m and an effective cross sectional area of 7.7×10-4 m2. Determine
the amount that each thighbone compresses under the extra weight.

55. 10.7 Elastic Deformation

56. Shear Deformation
It is possible to deform a solid object in a way other than stretching or compressing it.
When a top cover of a book is pushed the pages below it become shifted relative to the stationary bottom cover.
The resulting deformation is called a shear deformation.

57. Shear Deformation
When a force F is applied parallel to the surface of an object of area A, the object experiences a shear x (the top surface moves relative to the bottom surface) for an object with thickness L0.
The magnitude of the shearing force is proportional to the fractional shear x in thickness x/L0, rather than the absolute change x.

58. Shear Modulus
The constant of proportionality S is called the shear modulus.
Shear modulus depends on the nature of the material
For a given force, the material with higher S undergoes the smaller shear x .
Shear modulus has the units of pressure: N/m2

59. Shear Deformation And The Shear Modulus

60. SHEAR DEFORMATION AND THE SHEAR MODULUS
10.7 Elastic Deformation
SHEAR DEFORMATION AND THE SHEAR MODULUS
The shear modulus has the units of pressure: N/m2

61. 10.7 Elastic Deformation

62. Example 14: Shear Modulus of J-E-L-L-O
10.7 Elastic Deformation
Example 14: Shear Modulus of J-E-L-L-O
You push tangentially across the top
surface with a force of 0.45 N. The
top surface moves a distance of 6.0 mm
relative to the bottom surface. What is
the shear modulus of Jell-O?

63. Shear Modulus of J-E-L-L-O
10.7 Elastic Deformation
Shear Modulus of J-E-L-L-O
Jell-o can be deformed easily, because its Shear Modulus is
significantly ___________ than that of other rigid materials, e.g., steel

64. Young’s modulus vs. Shear modulus
Young’s modulus (Y) refers to the change in length of one dimension of a solid object as a result of tensile or compressive force.
The shear modulus (S) refers to a change in shape of a solid object as a result of shearing force.

65. Young’s modulus vs. Shear modulus
The tensile force is perpendicular to the surface
of area A
The shear force is parallel to the surface
The distances L and L0 (length) are parallel
The distances  x and L0 (thickness) are perpendicular

66. Volume Deformation
When applied compressive forces changes the size of every dimension (length, width, and depth), leading to a decrease in volume, the resulting deformation is called a volume deformation.
This kind of overall compression occurs when an object is submerged in a liquid.
The force acting in such situation s are applied perpendicular to every surface.
The perpendicular force per unit area is pressure.

67. Pressure
The pressure P is the magnitude of the force F acting perpendicular to a surface divided by the area A over which the surface acts:
SI unit of Pressure is Pa (pascal) = N/m2 F A

68. Volume Deformation
The change in pressure P needed to change the volume by an amount V is directly proportional to the fractional change V/V0 in volume:
Change in pressure P = final pressure P - initial pressure P0
Change in volume v = final volume V - initial volume V0

69. VOLUME DEFORMATION AND THE BULK MODULUS
10.7 Elastic Deformation
VOLUME DEFORMATION AND THE BULK MODULUS
B is the proportionality constant called Bulk modulus
The Bulk modulus has the units of pressure: N/m2
The negative sign represents the increase in pressure decreases the volume

70. 10.7 Elastic Deformation

71. Stress, Strain, and Hooke’s Law
All these modified equations specify the amount of force needed per unit area for a given amount of elastic deformation.
In general, the ratio of the magnitude of the force to the area is called the stress.
The right side of each equation involves the change in a quantity (L, X, or V) divided by a quantity (L or V) relative to which the change is compared. Each of these ratios are referred to as the strain that results from the stress.

72. Stress, Strain, and Hooke’s Law
Stress and strain are directly proportional to one another.
This relationship was first discovered by Robert Hooke and referred to a Hooke’s Law.

73. 10.8 Stress, Strain, and Hooke’s Law
In general the quantity F/A is called the stress.
The change in the quantity divided by that quantity is called the
strain:
HOOKE’S LAW FOR STRESS AND STRAIN
Stress is directly proportional to strain.
Strain is a unitless quantitiy.
SI Unit of Stress: N/m2

74. 10.8 Stress, Strain, and Hooke’s Law
In reality, materials obey Hooke’s law only up to a certain limit,
proportionality limit.
Elastic limit is the point beyond which the object no longer
returns to its original size and shape when the stress is removed.

75. Force A

76. Force A

77. Force A

78. Force A