Simple Harmonic Motion

Wenny Maulina

Simple harmonic motion

Simple harmonic motion (SHM)

• Solution:

)cos()( tAtx

• What is SHM?

A simple harmonic motion is the motionof an oscillating system which satisfiesthe following condition:

1. Motion is about an equilibrium position at which point no net force acts on the

system.2. The restoring force is proportional to and oppositely directed to the displacement.3. Motion is periodic.

t=0t=- /f w

Acosf

f= /(2 )w p

By Dr. Dan Russell, Kettering University

=w w0; =w 2w0 ; =w 3w0

Simple Harmonic Motion, SHMSimple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and 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

x F

Oscillations of a SpringHooke’s Law states Fs = -kx

Fs is the restoring force.It is always directed toward the equilibrium position.Therefore, it is always opposite the displacement from equilibrium.

k is the force (spring) constant.x is the displacement.

Oscillations of a Spring In a, the block is displaced to the right of x = 0.

The position is positive. The restoring force is directed to the left (negative).

In b, the block is at the equilibrium position.

x = 0 The spring is neither stretched nor

compressed. The force is 0.

In c, the block is displaced to the left of x = 0.

The position is negative. The restoring force is directed to the right (positive).

6

Example

A 0.42-kg block is attached to the end of a horizontal ideal

spring and rests on a frictionless surface. The block is pulled so

that the spring stretches by 2.1 cm relative to its length. When

the block is released, it moves with an acceleration of 9.0 m/s2.

What is the spring constant of the spring?

7

2.1cm

kx = ma

2/0.942.0100

1.2smk

mNk /1801001.2

0.942.0

Displacement in SHM

• Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left.

• The maximum displacement is called the amplitude A.

m

x = 0 x = +Ax = -A

x

Velocity in SHM

m

x = 0 x = +Ax = -A

v (+)

• Velocity is positive when moving to the right and negative when moving to the left.

• It is zero at the end points and a maximum at the midpoint.

v (-)

Acceleration in SHM

m

x = 0 x = +Ax = -A

• Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.)

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

+x-a

-x+a

F ma kx

Acceleration vs. Displacement

m

x = 0 x = +Ax = -A

x va

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

or

Note: Acceleration is always opposite to displacement.

F ma kx kx

am

Simple harmonic motion

Displacement, velocity and acceleration in SHM

• Displacement

)cos()( tAtx

• Velocity

)sin()(

)( tAdt

tdxtv

)cos()(

)( 2 tAdt

tdvta

• Acceleration

0

The diaphragm of a loudspeaker moves back and forth in simple

harmonic motion to create sound. The frequency of the motion is f

= 1.0 kHz and the amplitude is A = 0.20 mm.

(a)What is the maximum

speed of the diaphragm?

(b)Where in the motion does

this maximum speed

occur?

Example

(b) The speed of the diaphragm is zero when the diaphragm

momentarily comes to rest at either end of its motion: x = +A

and x = –A. Its maximum speed occurs midway between these

two positions, or at x = 0 m.

(a)

A loudspeaker diaphragm is

vibrating at a frequency of f =

1.0 kHz, and the amplitude of

the motion is A = 0.20 mm.

(a)What is the maximum

acceleration of the

diaphragm, and

(b)where does this maximum

acceleration occur?

Example

(b) the maximum acceleration occurs at x = +A and x = –A

(a)

The drawing shows plots

of the displacement x

versus the time t for three

objects undergoing simple

harmonic motion. Which

object, I, II, or III, has the

greatest maximum

velocity?

Example

The cone of a loudspeaker oscillates in

SHM at a frequency of 262 Hz. The

amplitude at the center of the cone is A =

1.5 x 10-4 m, and at t = 0, x = A. (a) What

equation describes the motion of the center

of the cone? (b) What are the velocity and

acceleration as a function of time? (c)

What is the position of the cone at t = 1.00

ms (= 1.00 x 10-3 s)?

Example

Solution:

.1650cosm105.1

)2cos()(

,2cos)0(

),cos()(

rad/s,1650Hz26222 a.

4 t

tAtx

AAx

tAtx

f

Solution:

m.102.1

s1000.1rad/s1650cosm105.1

ms,00.1at c.

1650cosm/s410)(

;1650sinm/s25.0)( b.

5

34

2

x

t

tdt

dvta

tdt

dxtv

A spring stretches 0.150 m when a 0.300-kg mass is gently attached to it. The spring is then set up horizontally with the 0.300-kg mass resting on a frictionless table. The mass is pushed so that the spring is compressed 0.100 m from the equilibrium point, and released from rest. Determine: (a) the spring stiffness constant k and angular frequency ω; (b) the amplitude of the horizontal oscillation A; (c) the magnitude of the maximum velocity vmax; (d) the magnitude of the maximum acceleration amax of the mass; (e) the period T and frequency f; (f) the displacement x as a function of time; and (g) the velocity at t = 0.150 s.

Example

Solution:

frequencyangular :ω

nt)displaceme (maximum amplitude:Am

kω whereωt,sin A x :solution simple a

kxdt

xdm

:Motion ofEquation

2

2

Oscillations of a Spring

F = - k x

k

x

m

F = - k x

x

Displacement : x = L θ

Returning force : F = - mg sin θ

Newton Law : F = m d2θ/dt2

Small oscillation approximation : sin θ θ

Acceleration : d2θ/dt2 = - (g/L) x

Solutions : Y = A sin wt , where w2 = g/L

Y = A cos wt

Y = A eiwt

Simple Pendulum

Taylor series :sin𝜃=𝜃−𝜃3

3 !+ 𝜃

5

5 !−⋯

• The fact that the velocity is zero at maximum displacement in simple harmonic motion and is a maximum at zero displacement illustrates the important concept of an exchange between kinetic and potential energy.

• If no energy is dissipated then all the potential energy becomes kinetic energy and vice versa, so that the values of (a) the total energy at any time, (b) the maximum potential energy and (c) the maximum kinetic energy will all be equal; that is

Energy in SHM

UKE

Energy in SHM

Energy conservation

UKE Energy conservation in a SHM

dx

dUkxFs

No friction22

2

1 ;

2

1kxUmvK

const. 2

1

2

1 22 kxmvE

EkxmvkA 222

2

1

2

1

2

1

22 xAm

kv

BTW:

2

0 2

1 kxdxFUU

x

s

w2

Energy in SHM

Energy conservation in a SHM

const. 2

1

2

1 22 kxmvE

ene

rgy

ene

rgy

distance from equilibrium pointTime

E

kinetic energy

potential energy

This graph shows the potential energy function of a spring. The total energy is constant.

Energy in SHM

An object of mass m = 0.200 kg that is vibrating on a horizontal frictionless table. The spring has a spring constant k = 545 N/m. It is stretched initially to x0 = 4.50 cm and then released from rest (see part A of the drawing). Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf = 2.25 cm and (b) xf = 0 cm.

Example

30

20

22

2

1

2

1

2

1kxkxmv ff

)( 220 ff xx

m

kv

0EE f

20

20

22

2

1

2

1

2

1

2

1kxmvkxmv ff

31

(a) Since x0 = 0.0450 m and xf = 0.0225 m,

(b) When x0 = 0.0450 m and xf = 0 m,

What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?

L

Example

What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?

2L

Tg

L

L = 0.993 m

Example

L

L

Many tall building have mass dampers, which are anti-sway devices to prevent them from oscillating in a wind. The device might be a block oscillating at the end of a spring and on a lubricated track. If the building sways, say eastward, the block also moves eastward but delayed enough so that when it finally moves, the building is then moving back westward. Thus, the motion of the oscillator is out of step with the motion of the building. Suppose that the block has mass m = 2.72 x 105 kg and is designed to oscillate at frequency f = 10.0 Hz and with amplitude xm = 20.0 cm.

(a) What is the total mechanical energy E of the spring-block system?

Example

Damped harmonic motion is harmonic motion with a frictional or drag force.

DAMPED OSCILLATION

There are systems in which damping is unwanted, such as clocks and watches.

Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.

The friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped.

The smallest degree of damping that completely eliminates the oscillations is termed “critical damping,” and the motion is said to be critically damped.

When the damping exceeds the critical value, the motion is said to be overdamped. In contrast, when the damping is less than the critical level, the motion is said to be underdamped.

DAMPED OSCILLATION

DAMPED OSCILLATION

When a periodically varying driving force with angular frequency is applied to a damped harmonic oscillator, the resulting motion is called a forced oscillation.

There are two frequencies involved in a forced oscillator:I. ω0, the natural angular frequency of the oscillator, without

the presence of any external force, andII. ω, the angular frequency of the applied external force.

If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.

Forced Oscillation

The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp.

Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.

Forced Oscillation

Please make a paper about Standing waves and Travelling waves

ASSIGNMENT 1Should Be Submitted Next Week