Oscillation

Question 1

What are oscillations and vibrations?

Answer 1

A type of periodic motion, motion repeats in a regular way as time passes.

Question 2

What is mechanical oscillation? What does it require?

Answer 2

When an object ‘oscillates’ - repeatedly moves backwards and forwards about an equilibrium position. Requires a resultant force always directed towards the position of equilibrium

Question 3

What is the restoring force?

Answer 3

The resultant force always directed towards the equilibrium position

Question 4

What is the displacement, x?

Answer 4

Distance and direction of the oscillating object from its equilibrium position

Question 5

What is SHM?

Answer 5

Repeated motion in a single plane, the restoring force is directly proportional to the displacement, and in the opposite direction

Question 6

What is the equation of a displacement vs time graph for SHM when t=0, x=A? What about for when t=0, x=0?

Answer 6

1) x = Acos(ωt)

2) x = Asin(ωt)

Question 7

What is the equation for angular frequency (ω)?

Answer 7

2πf, measured in rad/s

Question 8

How do you find the phase difference, for example between the displacement and velocity?

Answer 8

Determine the time that elapses between each quantity being at a maximum. The phase difference in terms of the fraction of a cycle can then be found by dividing the time that elapses by the time period. Conversions can be made by equation 1 full cycle to 360° or 2π radians

Question 9

What equation defines SHM? What does the equation show?

Answer 9

a = -ω²x

Acceleration is directly proportional to displacement but in the opposite direction

Question 10

What is the equation for the max value of acceleration?

Answer 10

a(max) = ω²A, since amplitude A is the maximum displacement

Question 11

What is the equation for the velocity of an object moving with SHM? Equation for max velocity and why?

Answer 11

1) v = ±2πf√ (A² - x²)

2) v(max) = 2πfA = ωΑ, because velocity is greatest at E.P. which corresponds to x = 0

Question 12

Describe the shape of a graph of acceleration vs displacement for an object oscillating with SHM

Answer 12

Straight line through the origin with gradient equal to -ω²

Question 13

Describe the shape of a graph of restoring force vs displacement for an object oscillating with SHM

Answer 13

A straight line through the origin with a gradient equal to -mω², where m is the mass of the object. This is because for constant mass, acceleration is directly proportional to the resultant force ( F=ma), the restoring force is directly proportional to the displacement but in the opposite direction

Question 14

What is the equation for the upwards resultant force in a mass-spring system?

Answer 14

F = kx, however since resultant force is upwards and displacement is downwards, it is written as F = -kx

Question 15

Using Newton’s second law, derive the equation for the acceleration of an oscillating mass in a mass-spring system. How does this equation conform to definition of SHM?

Answer 15

1) F = ma & F = -kx → ma = -kx → a = -kx/m

2) given that k/m is a constant, this shows that a is directly proportional to x but in the opposite direction

Question 16

Derive the formula for the time period in a mass-spring system

Answer 16

Compare SHM equation a = -ω²x with a = -kx/m shows that
ω² = k/m. Substituting ω = 2π/Τ into the equation gives
(2π/T)² = k/m which rearranges to give T = 2π√ (m/k)

Question 17

What does θ equal in a simple pendulum situation?

Answer 17

x/l

Question 18

What is the equation for the restoring force in a simple pendulum situation?

Answer 18

1) F = Wsinθ = mgsinθ, the small angle approximation states that for small angles (

Question 19

Using Newton’s second law, derive the equation for the acceleration of an oscillating mass in a simple pendulum system. How does this equation conform to definition of SHM?

Answer 19

1) F = ma, F = -mgx/l → a = -gx/l
2) this equation shows that the acceleration of an oscillating simple pendulum of constant length is directly proportional to and in the opposite direction to its displacement, in other words oscillates with SHM

Question 20

Derive the formula for the time period in a simple pendulum system

Answer 20

Compare SHM equation a = -ω²x with a = -gx/l shows that
ω² = g/l. Substituting ω = 2π/Τ into the equation gives
(2π/T)² = g/l which rearranges to give T = 2π√ (l/g)

Question 21

Give 3 examples of periodic motion

Answer 21

Swinging pendulums, masses oscillating on springs, atoms vibrating in a solid

Question 22

What is a free oscillation? What is the total energy of a free oscillation?

Answer 22

1) An idealised oscillation in which the amplitude of the oscillating object is constant, as no energy is input, and no energy is removed from the system by friction for example
2) The sum of the potential energy and the kinetic energy and is constant throughout the oscillation

Question 23

For a mass oscillating on a spring, what does the potential energy consist of?

Answer 23

The elastic strain energy in the spring, which is maximum at the bottom extreme of the oscillation and the GPE, which is maximum at the top extreme of the oscillation

Question 24

What is damping?

Answer 24

The removal of energy from an oscillating system, the extent of damping determines how long it takes for the oscillation to die away

Question 25

What is the difference between light damping and heavy damping?

Answer 25

If damping is light, the oscillations die away gradually. If the resistive forces are high, the system has heavy damping and the oscillated object moves slowly back into EP without oscillating

Question 26

What is critical damping? When would it be used?

Answer 26

1) When the resistive forces are just enough to prevent oscillation and the object returns to EP in the shortest possible time
2) mechanical systems to prevent vibration damage, front suspension on mountain bike

Question 27

What is a forced oscillation?

Answer 27

When energy is repeatedly transferred to the oscillation

Question 28

What is the driving frequency?

Answer 28

The frequency of the forced oscillations

Question 29

Give 3 examples of forced oscillation

Answer 29

1) someone being pushed on a swing
2) a person’s eardrum being forced to oscillate by sound waves
3) a loudspeaker being forced to oscillate by the electrical signal supplied to it

Question 30

In a forced oscillation, what does it mean when the driving frequency has the same value as the natural frequency? When will this stop happening?

Answer 30

1) The energy transfer to the oscillator occurs at maximum efficiency, causing the amplitude to of the forced oscillation to increase
2) A will continue to increase until the energy supplied by the driving system is equal to energy lost by damping

Question 31

What is resonance?

Answer 31

The effect of producing a large-amplitude oscillation by matching the driving frequency to the natural frequency

Question 32

When can resonance cause problems?

Answer 32

In mechanical systems e.g. bridges, tall buildings and machinery

Question 33

What are some useful applications of resonance?

Answer 33

1) key to operation of most musical instruments
2) microwave cooker
3) MRI scanning

Question 34

What is the resonant frequency?

Answer 34

The frequency at which resonance occurs

Question 35

What are Barton’s pendulums?

Answer 35

A way of demonstrating resonance. A heavy pendulum, the driver, is attached to a string and a series of lighter pendulums which differed in length. When the driver is set in motion, energy is transferred to the lighter pendulums and the begin to oscillate with the same frequency as the driver. Energy most efficiently transferred to the lighter pendulum with the same length and therefore natural frequency as the driver. This pendulum oscillates with large amplitude as it undergoes resonance with a phase difference of 90° or π/2 radians behind the driving pendulum.

Question 36

What determines the pitch of a note on a musical instrument?

Answer 36

The first harmonic. But higher harmonics can occur simultaneously with the first harmonic, and the number and amplitude of these determine the quality of the sound

Question 37

Why do the strings on musical instruments make almost no sound?

Answer 37

The thin wires cut through air very easily and cause very little vibration of the air molecules

Question 38

How does a guitar create sound?

Answer 38

The vibration of the string is transferred to the body of the guitar via the bridge. The body has been designed so that its own range of harmonic frequencies matches those of the strings, enabling the energy of vibration of the strings to be efficiently transferred to the body. theThe guitar body resonates and causes the air within it and surrounding it to vibrate, producing a travelling sound wave of sufficient amplitude to be heard

Question 39

What are the 3 key features to note from an amplitude of the driving system vs frequency of the driving oscillation graph?

Answer 39

1) with no or little damping, the largest value for the maximum amplitude occurs when the frequency of the driving oscillation equals the natural frequency
2) the resonant frequency decreases as the degree of damping increases
3) the resonance curve becomes less sharp as the damping is increased

Question 40

When does the resonant frequency equal the the natural frequency?

Answer 40

When a freely oscillation system occurs