Simple Harmonic Motion

In periodic motion, a body repeats a certain motion indefinitely, always returning to its starting point after a constant time interval and then starting a new cycle.

Every system that has this force exhibits a simple harmonic motion (SHM) and is called a simple harmonic oscillator.

Simple harmonic motion Simple harmonic motion is periodic motion in the is periodic motion in the absence of friction and produced by a absence of friction and produced by a restoring forcerestoring force that is directly proportional to the displacement and that is directly proportional to the displacement and oppositely directed.oppositely directed.

A restoring force, F, acts in the direction opposite the displacement of the oscillating body.

F = -kx

A restoring force, F, acts in the direction opposite the displacement of the oscillating body.

F = -kx

Hooke’s Law

F

x

m

A mass m attached to a spring executes SHM when the spring is pulled out and released. The spring’s PE becomes K as the mass begins to move, and the K of the mass becomes PE again as its momentum causes the spring to overshoot the equilibrium position and become compressed.

Displacement in SHM

m

x = 0 x = +Ax = -A

x

• The maximum displacement is called the amplitude A.

Velocity in SHM

m

x = 0x = 0 x = +Ax = +Ax = -Ax = -A

v (+)v (+)

• Velocity is Velocity is zero zero at the end points and a at the end points and a maximummaximum at the midpoint in either direction at the midpoint in either direction

v (-)v (-)

Acceleration in SHM

m

x = 0 x = +Ax = -A

• Acceleration is a maximum at the end points and it is zero at the center of oscillation.

+x-a

-x+a

F ma kx F ma kx

Acceleration vs. Displacement

m

x = 0 x = +Ax = -A

x va

Given the spring constant, the displacement, Given the spring constant, the displacement, and the mass, the and the mass, the accelerationacceleration can be found can be found from:from:

oror

Note: Acceleration is always Note: Acceleration is always oppositeopposite to displacement.to displacement.

F ma kx F ma kx kxa

m

kx

am

Conservation of Energy

The The total mechanical energy (total mechanical energy (PEPEss + K + K) ) of a SH of a SH

system is constant.system is constant.

m

x = 0 x = +Ax = -A

x va

For any two points A and B, we may write:For any two points A and B, we may write:

½mvA2 + ½kxA 2 = ½mvB

2 + ½kxB 2 ½mvA2 + ½kxA 2 = ½mvB

2 + ½kxB 2

A B

GRAPH OF SHMThe graph shown below depicts the up and down oscillation of the mass at the end of a spring. One complete cycle is from a to b, or from c to d.The time taken for one cycle is T, the period.

PERIOD AND FREQUENCY

The period T of a body of mass m attached to a spring of force constant k

UNITS:

T in seconds

f in Hz (s-1)

For small displacements a pendulum obeys SHM.

Its period is:

Simple Pendulum

The period and frequency DO NOT depend on the mass.

6.1 For the motion shown in the figure, find:a. Amplitudeb. Period c. Frequency

a. Amplitude: maximum displacement from equilibrium A = +- 0.75 cm

b. T = time for one complete cycle T = 0.2 s

c. f = 1/T = 1/0.2 = 5 Hz

6.2 A 200-g mass vibrates horizontally without friction at the end of a horizontal spring for which k = 7.0 N/m. The mass is displaced 5.0 cm from equilibrium and released. Find:a. Maximum speed

m = 0.2 kgk = 7 N/mxo = 0.05 m

2 2 21 1 1

2 2 2okx mv kx

vmax is at x = 0 then: 0

kv x

m 7

0.050.2

v = 0.295 m/s

COE

b. Speed when it is 3.0 cm from equilibrium.

x = 0.03 m 1

2

2 1

2

2 1

2

2kx mv kxo

2 2( )o

kv x x

m 2 27

0.05 0.030.2

v = 0.236 m/s

c. What is the acceleration in each of these cases?

F = ma = - kx

kxa

m

a. x = 0 therefore a = 0

b. x = 0.03 m therefore

7(0.03)

0.2a

= - 1.05 m/s2

6.3 As shown in the figure, a long, light piece of spring steel is clamped at its lower end and a 2.0-kg ball is fastened to its top end. A horizontal force of 8.0 N is required to displace the ball 20 cm to one side as shown. Assume the system to undergo SHM when released. Find: a. The force constant of the spring

F = 8 Nx = 0.2 mm = 2 kg

Fk

x 8

0.2 = 40 N/m

b. Find the period with which the ball will vibrate back and forth.

2m

Tk

2

240

= 1.4 s

SHM

6.3 In a laboratory experiment a student is given a stopwatch, a wooden bob, and a piece of cord. He is then asked to determine the acceleration of gravity. If he constructs a simple pendulum of length 1 m and measures the period to be 2 s, what value will he obtain for g?

L = 1 mT = 2 s

2L

Tg

2

2

4 Lg

T

2

2

4 (1)

2

= 9.86 m/s2

SHM

DAMPED HARMONIC MOTIONA system undergoing SHM will exhibit damping. Damping is the loss of mechanical energy as the amplitude of motion gradually decreases. In the mechanical systems studied in the previous sections, the losses are generally due to air resistance and internal friction and the energy is transformed into heat.For the amplitude of the motion to remain constant, it is necessary to add enough energy each second to offset the energy losses due to damping.

In many instances damping is a desired effect. For example, shock absorbers in a car remove unwanted vibration.

FORCED VIBRATIONS: RESONANCE An object subjected to an external oscillatory force tends to vibrate. The vibrations that result are called forced vibrations. These vibrations have the same frequency as the external force and not the natural frequency of the object.

If the external forced vibrations have the same frequency as the natural frequency of the object, the amplitude of vibration increases and the object exhibits resonance. The natural frequency (or frequencies) at which resonance occurs is called the resonant frequency.

EXAMPLES OF RESONANCE