# Oscillations notes

October 12, 2017 | Author: xwenhan | Category: Resonance, Oscillation, Force, Mechanics, Physics & Mathematics

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GCE ‘A’ Level H2 Physics (Syllabus 9745)

1

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Oscillations 1.

Simple harmonic motion An oscillatory motion in which the acceleration is always directed towards a fixed point (equilibrium position) and its magnitude is directly proportional to the distance from the fixed point. Its defining equation is 2 a = - ω x.

2.

Definition of some terms The displacement is the distance measured from the equilibrium point in a stated direction. The amplitude is the magnitude of the maximum displacement from the equilibrium position.

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GCE ‘A’ Level H2 Physics (Syllabus 9745)

2

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The period is the time taken to make one complete oscillation. The frequency is the number of oscillations completed per unit time. -1

The angular frequency is the angle by which the phase of the motion changes in unit time. It is expressed in rad s . Two particles are said to be oscillating in phase if the displacements at any instant in time are exactly similar, i.e. of the same direction and varying in the same manner. If the displacements at any instant are not exactly similar then there is a phase difference between them. Oscillations are said to be π rad out of phase if there is a phase difference of half a cycle. 3.

Variation of displacement, velocity and acceleration with time

x  x0 sin t t

v

dx  x0 cos t  v0 cos t dt

Note that magnitude of the maximum velocity v0 = x0 ω, and occurs at the equilibrium point.

t

a

Variation of velocity and acceleration with displacement

v   x0  x 2

a   2 x 5.

2

Energy in SHM There is a constant interchange of kinetic and potential energy. The total energy is always constant.

6.

Variation of kinetic energy and potential energy with displacement Substitute

1 1 v   x02  x 2 into Ek  mv 2 , we get Ek  m 2 ( x02  x 2 ) 2 2

Since total energy is constant and is equal to Ek when x = 0,

1 ET  m 2 x02 (horizontal straight line) 2

1 E p  ET  Ek  m 2 x 2 2 H2 Physics - Mr Tan Kheng Siang

2

Note that the magnitude of the maximum acceleration a0 = x0 ω and occurs at the maximum displacements and where velocity is zero.

t 4.

dv  v 0 sin t   x0 2 sin t  a 0 sin t dt

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GCE ‘A’ Level H2 Physics (Syllabus 9745) 7.

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Variation of kinetic energy and potential energy with time Substitute

v  x0 cos t

into

1 1 Ek  mv 2 , we get E k  m 2 x02 cos 2 t 2 2

At total energy is constant and is equal to maximum

kinetic

energy, E  1 m 2 x 2 (horizontal straight line) t 0 2

E p  ET  Ek 

1 m 2 x02 sin 2 t 2

Note that two cycles are completed in T. 8.

Free oscillations There are no resistive forces acting on the mass undergoing oscillation. The total energy does not decrease with time and the amplitude remains constant.

9.

Damped oscillations There are resistive forces acting on the mass undergoing oscillation. The total energy decreases with time as it is lost to the surroundings and the amplitude eventually decays to zero.

Light damping – Oscillations is maintained but amplitude decreases with time. Critical damping – No oscillations. The motion is brought to rest in the shortest possible time. Useful in car suspension systems and analogue meters. Over-damped system –No oscillations. The system returns very slowly to equilibrium position. 10. Forced oscillations A forced oscillation occurs when an external periodic force (driving force) acts on the oscillating system. The oscillating system oscillates with the frequency of the external force (driving frequency) and not its natural frequency. The amplitude of the forced oscillation depends on (i) the relative values of the natural frequency of the system and the value of the driving frequency and (ii) amount of damping in the system (see graph below). 11. Resonance A phenomenon when the amplitude of a forced oscillation becomes very large when the driving frequency is equal to the natural frequency of the oscillating system. This is because there is maximum transfer of energy from the driving force to the oscillating system.

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GCE ‘A’ Level H2 Physics (Syllabus 9745)

4

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Note the effects of damping on the frequency response curve – (i) maximum amplitude becomes lower and broader, (ii) maximum amplitude slightly shifted to the left. 12. Useful resonance and destructive resonance Make sure you are able to give examples of instances when resonance is useful (e.g. microwave heating involving water molecules, in musical instruments, magnetic resonance imaging) and when it is destructive (e.g. vibration of car parts at certain speeds, positive feedback in amplification systems giving a high-pitch squeal.)

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