3.2 Vibrations

Question 1

What is simple harmonic motion?

Answer 1

Simple harmonic motion (SHM) is a type of motion defined by a simple rule. In words, the acceleration of the body is directly
proportional to and in the opposite direction to its displacement from an equilibrium point.

Mathematically, we have:
a = -⍵2𝑥
The negative sign shows the opposite direction.

Question 2

What is the equation for the time period of a mass - spring simple harmonic system?

Answer 2

• T = 2𝜋√(m/k)
• m: kg k: Nm−1 T: s

Question 3

What is the equation for the time period of a simple harmonic pendulum?

Answer 3

T = 2𝜋√(L/g)
g: ms−2 l: m T: s

Question 4

Draw and explain a graph for the variation of acceleration with displacement during simple harmonic motion.

Answer 4

Question 5

What is the solution to the SHM equation? What is the velocity?

Answer 5

• 𝑥 = Acos(⍵t + ε)
• 𝑣 = -A⍵sin(⍵t + ε)

Question 6

What is the frequency?

Answer 6

Frequency is the number of oscillations a body makes per unit time (usually the second) and is given the symbol f.

Question 7

What is the time period?

Answer 7

The time period is the time that is taken for the body to make one oscillation and is given the symbol T. The relationship between frequency and period is simple. If the body makes f oscillations within one second then the time taken to complete each one is 1/f.

Question 8

What is the amplitude?

Answer 8

Amplitude is the maximum displacement from equilibrium that an object reaches. It is given the symbol A.

Question 9

What is phase?

Answer 9

Phase is the part that is added in the cos(ε) to make sure the mathematical description aligns with the physical situation. For example, if the body starts at its amplitude then ε = 0 but if the body starts at equilibrium then ε = pi/2 (as cos pi/2 =0).

Question 10

Draw the graph for potential energy and kinetic energy against displacement, for a SHM system.

Answer 10

Question 11

Using energy conservation, derive a formula for the velocity.

Answer 11

• In a mass-spring system, the kinetic energy is given by 1/2mv2. The potential energy is called elastic potential energy and it is stored in the spring when extended. When the spring is extended by x, the elastic potential energy is 1/2kx2 where k is the stiffness. Since the total energy must be the same we can say:
• 1⁄2 kA^2 = 1⁄2 mv^2 + 1⁄2 k𝑥^2
• v^2 = k/m(A^2-𝑥^2)
• v=±⍵ √A^2-𝑥^2

Question 12

Define free vibrations.

Answer 12

Free vibrations are oscillations that occur without an additional driving force. The frequency a system tends to vibrate at in a free vibration is called the natural frequency.

Question 13

Define forced vibrations.

Answer 13

A driving force causes the system to vibrate at a different frequency. For higher driving frequencies, the phase difference between the driver and the oscillations rises to π radians. For lower frequencies, the oscillations are in phase with the driving force. When resonance occurs, which is when it most efficiently transfers energy to the system, the phase difference will be π/2 radians.

Question 14

Define damping and explain what critical damping, overdamping and underdamping are.

Answer 14

• Damping occurs when an opposing force dissipates energy to the surroundings.
• Critical damping reduces the amplitude to zero in the quickest time.
• Overdamping is when the damping force is too strong and it returns to equilibrium slowly without oscillation.
• Underdamping is when the damping force is too weak and it oscillates with an exponentially decreasing amplitude.

Question 15

What happens to a vibration when damping is increased?

Answer 15

Question 16

What are some uses and implications of resonance in real life?

Answer 16

Uses include circuit tuning, microwave cooking etc.

Implications of resonance include that soldiers must break step when crossing bridges and vehicles must
be designed so there are no unwanted vibrations.