Simple Harmonic Motion

Question 1

What is an oscillator?

Answer 1

A mechanical system that is disturbed from equilibrium (a point of net zero force) by an applied force.

Question 2

What is SHM?

Answer 2

The oscillation, or back-and-forth movement, of a subject as a restoring force constantly acts opposite in direction to the displacement.

Question 3

Describe a horizontal mass-spring system in SHM.

Answer 3

A mass is displaced from equilibrium by an applied force which stretches it by a distance x.

Question 4

What is amplitude?

Answer 4

Amplitude is the maximum displacement from equilibrium

Question 5

What is frequency (SHM)? What is a period of oscillation? What is the relationship between them?

Answer 5

Frequency is the number of cycles per second while the period is the time (in s) for a cycle to complete. The two quantities are inverses of each other.

Question 6

How does mass affect periods/frequencies in a mass-spring system?

Answer 6

Higher mass leads to a longer period and thus, a smaller frequency. Lowering mass leads to shorter periods and higher frequencies.

Question 7

Change in spring constants causes? ( Increase )

Answer 7

Shorter relaxed length springs and higher frequencies.

Question 8

What is Angular Frequency?

Answer 8

The rate of change in angular displacement (like phase) per unit of time. Denoted by the symbol ω and is equal to 2πf

Question 9

What is phi and it’s meaning in SHM? What is something unique about phi and amplitude?

Answer 9

Phi is the symbol used to denote the initial phase of the system in an equation. Both the initial phase and amplitude are dependent on how the system is set into motion.

Question 10

What is uniform circular motion?

Answer 10

This is the motion of a subject going in a circle at a constant speed. The vertical component of UCM is akin to SHM.

Question 11

What is the symbol for phase difference? What is phase difference graphically represented?

Answer 11

The greek letter, ϕ (phi), is the symbol for phase difference.
A phase difference is a difference between two curves, typically one curve (a relative normal) with a phi equal to 0. A phase difference is measured in radians

Question 12

How much does the phase change by in one oscillation (in radians)?

Answer 12

2π

Question 13

How are the velocity and acceleration curves related to each other?

Answer 13

They are derivatives of the displacement curve.

Question 14

What are the phase differences between x(t), v(t) and a(t)?

Answer 14

x(t) and v(t) are out of phase by pi/2, v(t) is ahead. v(t) and a(t) are out of phase by pi/2 as well. The subsequent derivative curves are ahead of the last.

Question 15

When does a SHM system reach max velocity?

Answer 15

It occurs when the mass is reaching its equilibrium position.

Question 16

What is the relationship between displacement and acceleration curves?

Answer 16

The curves are typically “equal” and opposite. For example, a = 0 when d=0 since there are no forces acting at equilibrium.

Question 17

How does the restoring force affect a system?

Answer 17

The restoring force causes acceleration, which is also equal and opposite to the displacement.

Question 18

What are some relationships between kinetic energy and potential energy? Phase difference?

Answer 18

When kinetic energy is at its max, the potential energy is at zero. Similarly, the kinetic energy is at zero when the potential energy is at its max. They are always out of phase by pi.

Question 19

What do the energy curves look like on a graph dependent on time?

Answer 19

The kinetic energy and potential energy always add up to a sum, Total Energy, which is constant. The curves are always positive and only intercept when they each are half the sum, in ideal cases.

Question 20

What is the period of oscillation affected by?

Answer 20

It is affected by the mass and spring constant.

Question 21

A simple pendulum is a system where a mass swings back and forth undisrupted. How do mass, length and gravity affect the period length?

Answer 21

Mass does not affect the period length of the system.
Length is directly proportional to the period. Ex. if length increases, the period length increase.
Gravity is inversely proportional to the period.
Ex. if gravity increases, the period length decreases.

Question 22

A simple pendulum is a system where a mass swings back and forth undisrupted. How do mass, length and gravity affect the period length?

Answer 22

Mass does not affect the period length of the system.
Length is directly proportional to the period. Ex. if length increases, the period length increase.
Gravity is inversely proportional to the period.
Ex. if gravity increases, the period length decreases.

Question 23

What occurs in a system to create damped oscillators?

Answer 23

There’s a force present that leads to a loss in energy which opposes velocity.

Question 24

What occurs in a system to create damped oscillators?

Answer 24

There’s a force present that leads to a loss in energy which opposes velocity.

Question 25

What is the damping ratio?

Answer 25

It is a system boundary that is denoted by ζ (zeta). The value predicts how quickly a system’s oscillations will decay with time.

Question 26

What happens in a system under varying conditions? ζ > 1; ζ < 1 or ζ = 1

Answer 26

For ζ > 1, the system is overdamped
For ζ < 1, the system is underdamped
For ζ = 1, the system is critically damped

Question 27

Relationship between observed angular frequency and natural frequency.

Answer 27

The observed frequency is always less than the undamped system.

Question 28

Describe an overdamped system’s motion.

Answer 28

The mass when released will travel to equilibrium in a slow manner. It won’t oscillate.

Question 29

Describe an underdamped system’s motion.

Answer 29

The mass will oscillate with decaying amplitudes each time is passed it’s equilibrium position.

Question 30

Describe a critically damped system’s motion.

Answer 30

The mass will immediately return back to its equilibrium position in the quickest way possible.

Question 31

What is the steady-state solution?

Answer 31

It is when the damped oscillator was left for a long time to pass, so it is only driven by an external source.

Question 32

Relationship between ζ and the system’s resonant amplitude

Answer 32

As ζ gets smaller, in a system experiencing resonance, the higher the amplitude goes

Question 33

Describe Natural Frequency, Driven Frequency, Resonant Frequency and Damped Frequency.

Answer 33

Natural: the frequency observed with no damping or driving forces.
Driven: the frequency observed when a damped system has been oscillating for a long time.
Damped: the frequency observed when there are no driving forces.
Resonant: the frequency observed when a damped system will reach its max amplitude.