Topic 13 - Oscillations

Question 1

What is simple harmonic motion (SHM)?

Answer 1

A type of oscillation where the acceleration of an object is directly proportional to its displacement from the equilibrium position and acts in the opposite direction.

Question 2

What are these examples of?

• The pendulum of a clock
• A mass on a spring
• Guitar strings

Answer 2

Simple harmonic motion

Question 3

What does it meant for an objects oscillations to be periodic?

Answer 3

They are repeated in regular intervals of time according to their frequency or time period.

In other words, each complete cycle of oscillation takes the same amount of time

Question 4

What is the force that acts to return the object back to its equilibrium position, when moving in simple harmonic motion?

Answer 4

The restoring force

Question 5

If the displacement of an object in simple harmonic motion (SHM) is increased, what happens to the restoring force?

Answer 5

The restoring force increases

F ∝ x

Question 6

What is the Restoring Force equation for an object in Simple Harmonic Motion (SHM)?

Answer 6

F = -kx

Where:
* F = restoring force (N)
* k = a constant depending on the system (e.g. spring constant in Nm^-1)
* x = displacement from equilibrium position (m)

The negative sign shows that force, and therefore acceleration, will always act oppositely to the displacement (towards the centre of oscillation)

Question 7

Why is a person jumping on a trampoline NOT in Simple Harmonic Motion (SHM)?

Answer 7

Because the restoring force is not always directly proportional to displacement.

• In the air: The only force is weight (mg), which is constant and does not increase with distance from the trampoline mat (equilibrium).
• Contact only: The elastic restoring force from the trampoline only acts while the person is in contact with the mat, not during the whole motion.
• SHM requires: A restoring force that is proportional to displacement (F = -kx) and acts towards equilibrium at all points of the motion

Question 8

What is the equation to calculate the acceleration of an object in SHM?

Answer 8

a = -ω²x

Where:
* a = acceleration (ms^-2)
* ω = angular frequency (rad s^-1)
* x = displacement (m)

Question 9

What does the symbol ω represent in SHM?

Answer 9

ω is the angular frequency

It tells you how quickly the oscillation cycles in radians per second (rad s^-1)

Question 10

How is the symbol ω used differently in circular motion versus SHM?

Answer 10

• In circular motion, ω is the angular velocity, which is how quickly an object rotates
• In simple harmonic motion, ω is the angular frequency, which shows how quickly the phase of oscillation changes

Question 11

What is the relationship between the acceleration of an object oscillating in SHM and its displacement?

Answer 11

The acceleration of an object in SHM is directly proportional to the negative of its displacement from the equilibrium position

a ∝ -x

Question 12

What does the equation below demonstrate about acceleration and displacement in SHM?

a = -ω²x

Answer 12

The acceleration reaches its maximum value when the displacement is at a maximum (the amplitude)

Question 13

What can be said about an object in SHM when its displacement from equilibrium reaches a maximum value?

Answer 13

When an object in SHM has its displacement at a maximum value:
* This position is called the amplitude (A), which is the furthest distance from the equilibrium (centre) point
* At this point, the acceleration is also at its maximum (but in the direction back towards the centre)
* The speed of the object at maximum displacement is zero (it stops momentarily before reversing direction)

Question 14

When an object in SHM is at a maximum positive displacement, what can be said about the acceleration and speed of the object?

Answer 14

• Acceleration is at a maximum negative value (a = -a_max), acting in the opposite direction to displacement (points back towards equilibrium)
• Speed = 0 (the object momentarily stops before reversing direction)

Question 15

When an object in SHM is at a maximum positive acceleration, what can be said about the displacement and speed of the object?

Answer 15

• Displacement is at a maximum negative value (x = -A), which is in the opposite direction to acceleration
• Speed = (the object momentarily stops before reversing direction)

Question 16

When an object in SHM is at a maximum speed, what can be said about the displacement and acceleration of the object?

Answer 16

• x = 0
• a = 0

Question 17

What does the minus sign show in the equation below?

a = -ω²x

Answer 17

When the object is displaced to the right the direction of acceleration is to the left and vice versa (a and x are always in opposite directions to each other)

Question 18

What would the line for a graph of acceleration against displacement (from the equation a = -ω²x) for an object in SHM look like?

Answer 18

A straight line through the origin, but the line slopes downwards (similar to y = -x)

Question 19

What do the maximum and minimum displacement values for an acceleration-displacement graph of an object in simple harmonic motion show?

Answer 19

The points of amplitude of the oscillations (-A and +A)

Question 20

What does the gradient of an acceleration-displacement graph for an object in SHM give? (a/x)

Answer 20

The negative squared angular frequency

-ω²

Question 21

What is an equation for displacement in SHM?

Answer 21

x = A cos(ωt)

Where:
* x = displacement (m)
* A = amplitude (m)
* ω = angular frequency (rad s^-1)
* t = time (s)

Question 22

What must be true about the movement of an object in simple harmonic motion to use the equation below for displacement?

x = A cos(ωt)

Answer 22

This equation is used when an object is oscillating from its amplitude position (x = A or x = -A at t = 0).

The displacement will be at its maximum when cos(ωt) equals 1 or -1, when x = A

(ωt) = 0 radians (max positive) or π radians (max negative)

Question 23

What can the equation below for the displacement of an object in simple harmonic motion be used for?

x = A cos(ωt)

Answer 23

To find the position of an object in SHM with a particular angular frequency and amplitude at a moment in time

Question 24

How does the equation below for the displacement of an object in simple harmonic motion change if an object is oscillating from its equilibrium position (x = 0 at t = 0)?

x = A cos(ωt)

Answer 24

x = A sin(ωt)

This is because the sine graph starts at 0, whereas the cosine graph starts at a maximum

Question 25

What must be true about the movement of an object in simple harmonic motion to use the equation below for displacement? | x = A sin(ωt)

Answer 25

This equation is used when an object is oscillating from its **equilibrium** position (x = 0 at t = 0). The displacement will be at its maximum when sin(ωt) equals 1 or -1, when x = A (ωt) = π/2 radians

Question 26

What can be said about the consistency of the speed of an object in simple harmonic motion as it oscillates back and forth?

Answer 26

Its speed (magnitude of velocity) varies

Question 27

What equation shows how the speed of an object changes with the objects displacement x when oscillating in SHM?

Answer 27

v = ±ω√(A² - x²) Where: * v = speed (ms^-1) * A = amplitude (m) * ω = angular frequency (rad s^-1) * x = displacement (m)

Question 28

What does the equation below for how the speed of an object oscillating in SHM show about the relationship between amplitude and speed and why? | v = ±ω√(A² - x²)

Answer 28

The greater the amplitude A, the greater the speed v A ∝ v This is because when an oscillator (an object that is oscillating) has a greater amplitude, it has to travel a greater distance in the same time. Since d/t = speed, speed therefore is proportional to the amplitude

Question 29

What is meant by a simple pendulum?

Answer 29

An object attached to a fixed point above, swinging from side to side

Question 30

What equation can be used to calculate the **time period** of a **simple pendulum**?

Answer 30

T = 2π√(l/g) Where: * T = time period (s) * l = length of the pendulum swing (m) * g = the strength of gravity on the planet on which the pendulum is set up

Question 31

What is meant by a **mass-spring** system?

Answer 31

An object attached to the end of a spring that moves up and down due to a restoring force

Question 32

What equation can be used to calculate the **time period** of a **mass-spring** system?

Answer 32

T = 2π√(m/k) Where: * T = time period (s) * m = mass of the object on the end of the spring (kg) * k = spring constant of the material the system is made from (Nm^-1)

Question 33

How can an **experimental method** be used to observe the motion of a simple mass-spring system with a pencil and a mass?

Answer 33

1. Tie a pencil together with the mass 2. Attach the pencil-mass pair onto the end of a spring-mass system 3. Set the mass in free oscillations by displacing it downwards slightly 4. The oscillations will move the pencil up and down 5. On a piece of graph paper, allow the pencil to trace the path of the oscillations by pulling the paper sideways as the mass-spring system oscillates up and down 6. The oscillations will produce a curved, periodic graph, which will decrease in amplitude as the system slows down

Question 34

What graph can represent the **displacement** of an object in SHM?

Answer 34

A displacement-time graph

Question 35

What does it mean if a graph can be represented by **periodic functions**?

Answer 35

The graph can be described by sine and cosine curves

Question 36

What can all **undamped** SHM graphs be represented by?

Answer 36

**Periodic functions**

Question 37

What is meant by an **undamped** SHM graph?

Answer 37

A graph that shows the motion of an oscillating object where **no energy is lost to friction or air resistance** This means: * The **amplitude** of oscillations **stays the same** throughout * The shape of oscillation is a perfect **sine** or **cosine** curve, repeating identically for each cycle * The **motion continues indefinitely** with the same time period T and amplitude A

Question 38

What are the key features of a **displacement-time graph** for an **object oscillating** in simple harmonic motion?

Answer 38

* The **amplitude** of oscillations A can be found from the maximum value of displacement x * The **time period** of oscillations T can be found from reading the time taken for one full cycle * The graph might not always start at 0 * If the oscillations start at the positive or negative amplitude, the displacement will be at its maximum

Question 39

What graph can represent the **velocity** of an object in SHM?

Answer 39

A velocity-time graph

Question 40

What are the key features of a **velocity-time graph** for an **object oscillating** in simple harmonic motion?

Answer 40

* It is 90º out of phase with the **displacement-time graph** * The velocity of an oscillator at any time can be determined from the **gradient** of the **displacement-time graph** (v = ∆x/∆t) * The velocity is at its **maximum** when the objects displacement is zero

Question 41

What is meant by a **forced oscillation**?

Answer 41

Where oscillations are acted on by a periodic external force, where energy is given in order to sustain oscillations

Question 42

What is meant by the **driving frequency**?

Answer 42

The frequency of the applied (external) force that causes forced oscillations in a system

Question 43

What is meant by the **natural frequency**?

Answer 43

The frequency at which an oscillating system vibrates freely, with no external force acting on it.

Question 44

What happens in an **oscillating system** when the **driving frequency** approaches the **natural frequency** of the oscillator?

Answer 44

The system gains more energy from the driving force, and so the amplitude of oscillation increases

Question 45

What happens in an **oscillating system** when the **driving frequency** becomes equal to the **natural frequency** of the oscillator?

Answer 45

The system **gains energy** most efficiently from the driving force and **vibrates** with its **maximum amplitude** (this is **resonance**)

Question 46

What is resonance?

Answer 46

Resonance is when the frequency of the applied force (**driving force**) to an oscillating system is equal to its **natural frequency**. Therefore the **amplitude** of the **resulting** oscillations **increases** significantly.

Question 47

What can be said about the kinetic energy of an oscillating system at resonance?

Answer 47

At resonance, the system transfers the maximum kinetic energy possible.

Question 48

# Think oscillating systems What is **damping**?

Answer 48

The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system

Question 49

Why do all oscillators, in practice, eventually stop oscillating?

Answer 49

**Resistive forces**, such as friction or air resistance, act in the **opposite direction** to the **motion** of an oscillator. These forces cause the **amplitude** and **energy** of the oscillations to **decrease** over time, causing the oscillator to stop.

Question 50

What causes **damping** on an oscillating simple harmonic system?

Answer 50

Resistive forces (e.g. air resistance, friction etc.)

Question 51

In SHM, what does not change as the **amplitude** of oscillations **decreases** due to resistive forces?

Answer 51

The frequency of the oscillations (and therefore the time period also stays the same), even as the amplitude decreases.

Question 52

What are the three types of damping on an oscillating system?

Answer 52

* Light damping * Critical damping * Heavy damping

Question 53

What is meant by **light damping** on an oscillating system?

Answer 53

The **amplitude** of oscillations **decays exponentially** with time

Question 54

What is meant by **critical damping** on an oscillating system?

Answer 54

Where the oscillator will **return to rest** (at its equilibrium position) in the **shortest possible time** (with more-or-less no oscillations)

Question 55

What is meant by **heavy damping** on an oscillating system?

Answer 55

Where the oscillator will **return to rest** (at its equilibrium position) over a **long period of time** (with more-or-less no oscillations)

Question 56

What is a **free oscillation**?

Answer 56

An oscillation where there are **only internal forces** (no external forces) acting and there is **no energy input**

Question 57

What is meant by a resonance curve?

Answer 57

The curve produced on a graph of driving frequency f against amplitude A of oscillations

Question 58

What are they key features of a **resonant curve** graph

Answer 58

- When the driving frequency (f) < natural frequency (f₀), the amplitude of oscillation increases - At the peak where f = f₀, the amplitude is at its maximum (which is called resonance) - When f > f₀, the amplitude of oscillations starts to decrease

Question 59

When the degree of damping is increased on an oscillating object, how is the **resonance graph* altered?

Answer 59

- The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers - The resonance peak broadens - The resonance peak moves slightly to the **left** of the natural frequency when heavily damped Therefore, damping **reduced** the sharpness of resonance and **reduces** the amplitude at resonant frequency

Question 60

What is the relationship between the **damping** and **amplitude** of an oscillator?

Answer 60

They are inversely proportional to each other damping ∝ 1/amplitude

Question 61

Why can the amplitude of oscillations be reduced due to the **plastic deformation** of a **ductile** material?

Answer 61

The kinetic energy of the oscillator is reduced and transferred into the deformation of the material (e.g. a climbing rope provides critical damping by immediately stopping the climber from bouncing)