Chapter 18: Simple Harmonic Motion

Question 1

what is the equilibrium position in SHM?

Answer 1

The lowest point because the system will always try to return to that point.

Question 2

What is oscillatory motion? give examples

Answer 2

A oscillating motion is when an object moves repeatedly one way then in the opposite direction past its equilibrium position.

• up and down spring
• side to side pendulum
• small boat moving side to side

Question 3

What happens to the displacement of the object from equilibrium in SHM? Describe it

Answer 3

It continually changes since it keeps moving after released from no equilibrium postion.

• It decreases as it returns to equilibrium before reversing and increasing as it moves away from equilibrium then decreases as it returns to equilibrium then increasing as it moves away from equilibrium towards its starting postion

Question 4

What is amplitude?

Answer 4

The maximum displacement d the oscillating object from equilibrium. If amplitude is constant and no frictional force is present, the oscillations are described as free vibrations

Question 5

What is time period?

Answer 5

it is the time for one complete cycle of oscillation ( object passes through the same position in same direction)

T = 1/f

Question 6

What is frequency?

Answer 6

The number of cycles per second made by an oscillating object measured in Hertz(Hz)

Question 7

What is angular frequency?

Answer 7

Angular frequency w = 2π/T in radian per second (rads-1)

Question 8

What is phase difference

Answer 8

Two objects in the same time period with the same oscillating motion a few seconds apart = ∆t/T

phase difference in radians =
2π∆t/T

where t is time between successive instants when two objects are at maximum displacement in the same direction

Question 9

Describe the change in velocity of an object in SHM

Answer 9

The object speeds up as it return to equilibrium position and slows down as it moves away from equilibrium.

Question 10

If friction is negligible what happens to the amplitude in SHM?

Answer 10

It remains constant

Question 11

What is the gradient of a displacement time graph in SHM? When is it greatest and when is it zero?

Answer 11

Variation of velocity.

The magnitude of the velocity is greatest when the displacement - time graph is greatest. ( at zero displacement when the object passes through equilibrium.)

It is zero when the gradient of the displacement time graph is zero (maximum displacement in either direction)

Question 12

What is the gradient of a velocity-time graph? When is the magnitude greatest and when is it zero?

Answer 12

The change in acceleration with time

It is greatest when gradient is greatest, when velocity is zero and occurs at maximum displacement in the opposite direction.

It is zero when the gradient of velocity time graph is zero, when displacement is zero and velocity is maximum

Acceleration is always in the opposite direction to the displacement.

Question 13

What is simple harmonic Motion?

Answer 13

An oscillating Motion where the acceleration is proportional to the displacement and always in the opposite direction to the displacement (towards equilibrium)

a ∝-x

If displacement is negative, acceleration is positive

Acceleration a = -w^2x

Question 14

How can displacement of an object from equilibrium position position be calculated?

Answer 14

X = Acos (wt)

Where amplitude A = r

And θ is the angle the ball moves through from its position when x = A

At time t after the object passes through this position

θ = 2πt/T =wt

Question 15

What direction is the resultant force on an oscillating object?

Answer 15

The resultant force acts towards the equilibrium position and is described as the restoring force. It is proportional to displacement meaning acceleration is also proportional to displacement and always acts towards equilibrium

Question 16

Why does an object oscillate with simple harmonic Motion?

Answer 16

As long as the restoring for is proportional to the displacement from equilibrium, the acceleration is proportional to the displacement and always acts towards equilibrium causing the object to oscillate with SHM

Question 17

What determines the frequency of oscillation of a loaded spring?

Answer 17

Frequency of oscillation of a trolley can be changed by loading the trolley with extra mass or replacing the springs with different stiffness

Frequency is reduced by adding extra mass and using weaker springs

Question 18

What effect does adding extra mass have on the frequency of oscillation?

Answer 18

It reduces the frequency.

Adding extra mass increases the inertia of the system, at a given displacement the trolley would then be slower than if the extra mass had not been added and the cycle of oscillation would take longer.

Question 19

What effect does using weaker springs have of the frequency of oscillations of a loaded spring.

Answer 19

The frequency will be reduced, as the restoring force on the trolley at any given displacement will be less so it’s speed and acceleration will also be less and each cycle oscillation will take longer.

Question 20

What is the equation that shows the relationship between mass and spring constant and how it affects frequency?

Answer 20

Assuming spring obeys hookes law Ts is proportional to ΔL

Ts = kΔL

Change in tension from equilibrium
ΔTs = -kx

restoring force = -kx

therefore

a = restoring force/mass = -kx/m

Since a = -W^2x where w^2 = k/m

A = -w^2/x

Question 21

How can time period be calculated in relation to springs?

Answer 21

T = 2 π √m/k or 2 π √L/g

Question 22

How can tension be calculated in SHM

Answer 22

Ts - mg = mv^2/L where T is the tension of the rope acting directly upwards.

Question 23

Explain the theory of a simple pendulum taking into account its horizontal and vertical components.

Answer 23

If the object is displaced from equilibrium then released, it oscillates about the lowest point at displacement s. When θ is is at an angle to the vertical, the weight mg has components.

Mgcos θ is perpendicular to the path of the object

Mgsin θ is along the path towards the equilibrium position

The restoring force F = -mg sin θ

Therefore a =F/m = -mgsin θ /m = -gsin θ

As long as θ does not exceed approximately 10 degrees then sin θ = s/L

Therefore a = -g/L(s) = -w^2s where w^2 = g/L

So the object oscillates with simple harmonic Motion because it’s acceleration is proportional to the displacement from equilibrium and always acts towards equilibrium

Question 24

When is an object freely oscillating?

Answer 24

An object is freely oscillating when it is oscillating at a constant amplitude because there is no friction acting on it. If friction was present amplication would eventually decrease until the oscillations cease

Question 25

Describe the energy of a system as it changes from potential to kinetic energy if friction is absent.

Answer 25

If friction is absent the total energy of the system is constant and is equal to is maximum potential energy.

The Ep changes with displacement x from equilibrium using Ep = 1/2kx^2 where k is spring constant

Et is therefore 1/2kA^2 where A is amplitude of the spring

Since the total energy Et = Ek + Ep

Then Ek = Et - Ep or 1/2k(A^2 - x^2)

Question 26

What is the simple harmonic speed equation?

Answer 26

Using Ek=1/2mv^2 gives 1/2mv^2 = 1/2k(A^2 - x^2) where v is the speed of the object at displacement x

Since w^2 = k/m

The equation can be written v^2 = w^2 (A^2 - X^2)

V = w √(A^2 - X^2)

If x is 0 it gives the maximum speed wA

Question 27

Describe what happens in a energy - displacement graph

Answer 27

The potential energy curve is parabolic, given by Ep = 1/2kx^2

While the kinetic energy curve is an inverted parabola given by

Ek = Et - Ep or 1/2k(A^2 - x^2)

Question 28

What is the potential energy at maximum displacement? Kinetic energy at 0 displacement?

Answer 28

1/2KA^2 which is the same as kinetic energy at zero displacement

Question 29

Why do oscillations in a simple pendulum gradually stop?

Answer 29

This is due to air resistance which gradually reduces the total energy of the system. Where friction on air resistance is present the amplitude decreases

Question 30

What is a dissipative force?

Answer 30

Forces that cause amplitude of the system to decrease and dissipate the system’s energy to the surrounding area as thermal energy.

Question 31

When is motion said to be damped?

Answer 31

When dissipative forces which affect the amplitude and systems energy are present.

Question 32

What is light damping?

Answer 32

Light damping occurs when the time period is independent of the amplitude, so each cycle takes the same length of time as the oscillations die away. the amplitude reduces by the same fraction each cycle.

Question 33

What is critical damping?

Answer 33

Just enough to stop the system oscillating after it has been displaced from equilibrium and released. It return to equilibrium in the shortest possible time without over shooting if damping is critical. in respect of a displacement time graoh critical damping appears as a sharp downwards line.

Question 34

What is heavy damping

Answer 34

When the damping is so strong that the displaced object returns to equilibrium much more slowly than critical damping. No oscillating motion occurs. Appears as a stretched-out descending line on a displacement-time graph.

Question 35

What is periodic force?

Answer 35

A force applied at regular intervals. pushing someone on a swing at regular intervals.

Question 36

What is natural frequency?

Answer 36

When the sytem oscillates without a periodic force being applied, its frequency is natural frequency

Question 37

What is forced vibration/oscillation?

Answer 37

When a periodic force is applied to an oscillating system.

Question 38

What is the phase difference between displacement and periodic force in a system oscillating at maximum amplitude?

Answer 38

1/2π

Question 39

What is resonance?

Answer 39

When the periodic force is exactly in phase with the velocity of the oscillating system

Question 40

What is resonant frequency?

Answer 40

Frequency at maximum amplitude

Question 41

What is the relationship between damping and resonance?

Answer 41

The lighter the damping the larger the maximum amplitude becomes at resonance and the closer the resonant frequency of the system

The resonant curve on an amplitude-frequency graph will therefore be sharper

Question 42

What happens when the appliedd frequency becomes increasingly larger than the resonant frequency of mass-spring system?

Answer 42

• The amplitude will decrease more and more

-Phase difference between the displacement and the periodic force increases from 1/2π until displacement is π radians out of phase with the periodic force

Question 43

What happens when the oscillating system with little or no damping is at resonance?

Answer 43

The applied frequency of the periodic force = the natural frequency of the system

Question 44

What is the condition for simple harmonic motion

Answer 44

-Acceleration acts in the opposite direction to the displacement
- Acceleration is proportional to displacement

Question 45

Describe how the velocity of a bungee jumper changes during a jump
from the moment he jumps off the starting platform to the next time he
reaches the highest point of his jump

Answer 45

the velocity of a bungee jumper changes continuously throughout the jump, starting from zero at the beginning, reaching a maximum during free fall, decreasing as the bungee cord engages and eventually reverses the motion, and finally increasing again in a negative value as the jumper ascends towards the highest point of their jump.

Question 46

A metre rule is clamped to a table so that part of its length projects at
right angles from the edge of the table, as shown in Figure 3.A 100g
mass is attached to the free end of the rule. When the free end of the rule
is depressed downwards then released, the mass oscillates. Describe
how you would find out if the oscillations of the mass are free oscillations.

Answer 46

We would observe the amplitude of the mass, iif it remains constant it would mean that there are no frictional forces slowing it down suct as friction or air resistance therefore it is a free oscillation. As long as friction is negligible, amplitude remains constant

Question 47

What is the solution for displacement on a sine wave?

Answer 47

x = Asin(wt + Φ)

Where Φ is phase difference between instants when t = 0 and x = 0

Question 48

What are the conditions for simple harmonic motion?

Answer 48

Acceleration acts in the opposite direction to the displacement. ( Always directed towards equilibrium)

Acceleration is directly proportional to displacement

Question 49

When should you use Asin(wt) as opposed to Acos(wt)

Answer 49

Asin(wt) when t = 0 and the oscillator is at equilibrium.

While Acos(wt) when t = 0 and the oscillator is at amplitude.

Question 50

What is natural frequency?

Answer 50

Frequency at which an object will oscillate after initial disturbance.

Question 51

When does resonance occur in relation to natural frequency?

Answer 51

If the frequency of the driving force matches the natural frequency, resonance occurs. The amplitude increases considerably.

Question 52

What is a resistive force and what effect does it have?

Answer 52

A resistive force is a force acting in a direction which opposes the velocity of the object.

It reduces the amplitude of the object effectively slowing the object down.

The peak amplitude occurs at a lower frequncy

Question 53

What is a resonant force and what effect does it have?

Answer 53

A resonant force is a force acting in the same direction as the velocity of the object. It acts in the same frequency as the objects natural frequency.

It increases the amplitude of the object, effectively speeding the object up.