Oscillations

Question 1

Free Oscillations: Natural frequency

Answer 1

The frequency at which a body vibrates when there is no (resultant force) resistive force acting on it.
i.e. t. The guitar string vibrates at a particular frequency (the number of vibrations per unit time). This is called its natural frequency of vibration, and it gives rise to the particular note that you hear. Change the length of the string, and you change the natural frequency.

Question 2

Forced Oscillations

Answer 2

Vibrations from the bus engine are transmitted to your body, causing you to vibrate with the same frequency. These are not free vibrations of your body; they are forced vibrations. Their frequency is not the natural frequency of vibration of your body, but the forcing frequency of the bus.

Question 3

Limit of observing frequency by eyes

Answer 3

The eyes cannot respond rapidly enough if the frequency of oscillation is more than about 5 Hz (five oscillations per second); anything faster than this appears as a blur.

Question 4

Explain a general oscillation (no damping)

Answer 4

A trolley accelerates as it moves towards the centre of the oscillation. It is moving fastest at the centre. It decelerates as it moves towards the end of the oscillation. At the extreme position, it stops momentarily, reverses its direction and accelerates back towards the centre again.

Question 5

Amplitude, x○

Answer 5

The maximum displacement from the equilibrium position is called the amplitude x○ of the oscillation.

Question 6

Period, T

Answer 6

The period of an oscillating system is the time taken to make one complete oscillation.

Question 7

Phase and Phase difference

Answer 7

Phase : the point an oscillating particle has reached within the complete cycle of an oscillation.
Phase difference : the difference in the phases of two oscillating particles measured in degrees or radians

Question 8

How to calculate Phase difference

Answer 8

STEP 1 : Measure the time interval t between two corresponding points on the graphs.
STEP 2 : Determine the period T for one complete oscillation.
STEP 3 : Calculate the phase difference as a fraction (phase diff = t / T) of an oscillation.
STEP 4 : Convert to degrees and radians. There are 360° and 2π rad in one oscillation.

Question 9

Simple harmonic motion

Answer 9

A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position, and in the opposite direction to its displacement.

Question 10

Examples for SHM (basic)

Answer 10

1. When a pure (single tone) sound wave travels through air, the molecules of the air vibrate with s.h.m.
2. When an alternating current flows in a wire, the electrons in the wire vibrate with s.h.m.
3. There is a small alternating electric current in a radio or television aerial when it is tuned to a signal
   in the form of electrons moving with s.h.m.
4. The atoms that make up a molecule vibrate with s.h.m.

Question 11

Three requirements for SHM of a mechanical system

Answer 11

• A mass that oscillates
• A position where the mass is in equilibrium
• A restoring force that acts to return the mass to its equilibrium position; the restoring force F is
  directly proportional to the displacement x of the mass from its equilibrium position and is directed
  towards that point.

Question 12

Changes of velocity in SHM

Answer 12

(Pendulum for instance) As it swings from right to left its velocity is negative. It accelerates towards the equilibrium position and then decelerates as it approaches the other end of the oscillation. It has positive velocity as it swings back from left to right. Again, it has maximum speed as it travels through the equilibrium position and decelerates as it swings up to its starting position.

Question 13

Velocity-Time graph

Answer 13

When time t = 0, the mass is at the equilibrium position and this is where it is moving fastest. Hence, the velocity has its maximum value at this point. Its value is positive because at time t = 0 it is moving towards the right.
Fig. 18.15 on pg # 372

Question 14

Acceleration-Time graph

Answer 14

At the start of the oscillation, the mass is at its equilibrium position. There is no
resultant force acting on it so its acceleration is zero. As it moves to the right, the restoring force acts
towards the left, giving it a negative acceleration. The acceleration has its greatest value when the mass
is displaced farthest from the equilibrium position. Whenever the mass has a positive displacement (to the right), its acceleration is to the left, and vice versa.
“acceleration ∝ −displacement”

Fig. 18.15 on pg # 372

Question 15

Angular frequency

Answer 15

An object moves round through 2π radians. The phase of the oscillation changes by 2π rad during one oscillation. Hence, if there are f oscillations in unit time, there must be 2πf radians in unit time. This quantity is the angular frequency of the s.h.m.
ω = 2 π f = 2 π / T
UNIT: rad s-1

Question 16

Equations of SHM

Answer 16

x = x○ cos ω t
(graph starts at equil. when t = 0 sec)
x = x○ sin ω t
(graph start at max displacement when t = 0 sec)

p.s. we use the cosine function to represent the velocity since it has maximum value when t = 0.

Question 17

Acceleration and displacement equation + relevance

Answer 17

a = −ω2x
The greater the displacement x, the greater the acceleration a. The minus sign shows that, when the object is displaced to the right, the direction of its acceleration is to the left.

Question 18

Why does angular frequency comes in the equation for acceleration, a = −ω2x

Answer 18

i.e. A mass hanging on
a spring, so that it can vibrate up and down. If the spring is stiff, the force on the mass will be greater; it will be accelerated more for a given displacement and its frequency of oscillation will be higher.

Question 19

Purpose of minus sign in acceleration equation, a = −ω^2x

Answer 19

If a and x were in the same direction (no minus sign), the body’s acceleration would increase as it moved away from the fixed point and it would move away faster and faster, never to return.

Question 20

RADIAN MODE on calculator

Answer 20

Shift and Control and PRESS 2 ‘Degree Unit’ to change Degrees into Radians
p.s. Answer will be completely wrong otherwise.

Question 21

Equation to deduce speed of an oscillator at any point in an oscillation, including its maximum speed.

Answer 21

v = ± √ X○ ^2 − x ^2
(max velocity after simplifying) v○ = ω x○

Question 22

Why is v○ ∝ x○ (from the equation v○ = ω x○)

Answer 22

A simple harmonic oscillator has a period that is independent of the amplitude. A greater amplitude means that the oscillator has to travel a greater distance in the same time–hence it has a greater speed. The equation also shows that the maximum speed is proportional to the frequency. Increasing the frequency means a shorter period. A given distance is covered in a shorter time–hence it has a greater speed.

Question 23

Energy graphs

Answer 23

The graph shows that:
⁍ kinetic energy is maximum when displacement x=0
⁍ potential energy is maximum when x = ±x○
⁍ at any point on this graph, the total energy (k.e. + p.e.) has the same value.

Question 24

Damping

Answer 24

A damped oscillation is an oscillation in which resistive forces cause the energy of the system to be transferred to the surroundings as internal energy. The decrease in amplitude is exponential: great decrease followed by less decrease until slowly levels off.
p.s. The frequency of the oscillation does not change as the amplitude decreases.

Question 25

Energy and damping

Answer 25

In an unchanged oscillation, the total energy of the oscillation remains constant. There is a regular interchange between potential and kinetic energy. By introducing friction, damping has the effect of removing energy from the oscillating system, and the amplitude and maximum speed of the oscillation decrease.

Question 26

Resonance

Answer 26

Resonance occurs when the frequency of the driving force is equal to the natural frequency of the oscillating system. the system absorbs the maximum energy from the driver and has maximum amplitude.
(Must read Practical Activity 18.3, pg # 381, 382.)

Question 27

Conditions that apply to Resonating systems

Answer 27

• Natural frequency is equal to the frequency of the driver.
• Its amplitude is maximum.
• It absorbs the greatest possible energy from the driver.

Question 28

Critical damping (+ above and below) explanation

Answer 28

The minimum damping that causes the oscillating system to return to its equilibrium position in the minimum time and without oscillating. Any lighter damping will allow the system to oscillate one or more times any heavier damping will cause the system to take a longer time to return to its equilibrium position.
Under-damping results in unwanted oscillations; over-damping results in a slower return to equilibrium.
p.s. must look at graphs of damping in textbook pg # 383, 384.`

Question 29

Examples of Resonance

Answer 29

Textbook pg # 384
Example 1 : microwave cooking.
The microwaves used have a frequency that matches the natural frequency of vibration of water molecules (the microwave is the ‘driver’ and the molecule is the ‘resonating system’). The water molecules in the food are forced to vibrate and they absorb the energy of the microwave radiation. The
water gets hotter and the absorbed energy spreads through the food and cooks or heats it.
Example 2 : Magnetic resonance imaging (MRI) is used in medicine to produce images such as Figure 18.36, showing aspects of a patient’s internal organs. Radio waves having a range of frequencies are used, and particular frequencies are absorbed by particular atomic nuclei. The frequency absorbed depends on the type of nucleus and on its surroundings. By analysing the absorption of the radio waves, a computer-generated image can be produced.
Example 3 : A radio or television also depends on resonance for its tuning circuitry. The aerial picks up signals of
many different frequencies from many transmitters. The tuner can be adjusted to resonate at the frequency of the transmitting station you are interested in, and the circuit produces a large-amplitude
signal for this frequency only.