Further Mechanics

Question 1

Describe motion in a circular path at a constant speed

Answer 1

Motion in a circular path at constant speed implies there is an acceleration and requires a centripetal force

Question 2

Which direction does a Centripetal Force act

Answer 2

A centripetal force always acts towards the centre of the circle

Question 3

What is Angular Speed

Answer 3

The angle an object moves through per unit time

Question 4

What are the Equations for Angular Speed

Answer 4

ω = v / r

Angular Speed = Velocity / Distance

ω= 2πf

Angular Speed = 2π x Frequecny

Question 5

What are the Equations for Centripetal Acceleration

Answer 5

a = v² / r = ω²r

Acceleration = Velocity² / Distance = Angular Speed² X Distance

Question 6

What are the Equations for Centripetal Force

Answer 6

F = mv² / r = mω²r

Centripetal Force = Mass x Velocity² / Distance = Mass x Angular Speed² x Distance

Question 7

When is an object experencing simple harmonic motion

Answer 7

When its acceleration is directly proportional to displacement from the equilibrium point and is in the opposite direction

Question 8

What is the equation for accleration in simple harmonic motion

Answer 8

a = -ω²x

Accleration = - Angular Speed² x Displacement

Question 9

What is the equation for displacement in simple harmonic motion

Answer 9

x = A Cos(ωt)

Displacement = Amplitude x Cos(Angular Speed x Time)

Question 10

What is the equation for velocity in simple harmonic motion

Answer 10

v = ± ± ω √(A² - x²)

Velocity = ± Angular Speed √(Amplitude² - Displacement²)

Question 11

When are velocity and acceleration at their maximum and minimum on a pendulm

Answer 11

• Velocity Max = Centre
• Velcoity Zero = Amplitude
• Acceleration Max = Amplitude
• Accleration Zero = Centre

Question 12

What are the equations for velocity max and acceleration max

Answer 12

• V_max = ωA
• a_{max =} ω²A

Question 13

Describe the displacement-time graph of simple harmonic motion

Answer 13

Question 14

Describe the Velocity-Time Graph of Simple Harmonic Motion

Answer 14

Question 15

Describe the Acceleration-Time Graph of Simple Harmonic Motion

Answer 15

Question 16

Describe the Time Period Equation for a Simple Harmonic Pendulum

Answer 16

T = 2π √(L / g)

Time Period = 2π √ (Length of the String / Acceleration due to Gravity)

Question 17

Explain why the simple harmonic pendulum only works for small angles

Answer 17

• The angle by which the pendulum is displaced must be less than 10°
• This is because during the derivation of the above formula a small angle approximation is used, and so for larger initial angles this approximation is no longer valid, and would not be a good model
• Therefore the amplitude must be small

Question 18

What are the Limitations of the Simple Harmonic Motion Equations for a Simple Pendulum

Answer 18

• The amplitude is small
• The string is inextensible
• The bob used is a point mass

Question 19

How would you measure the time period of the oscillations of a simple pendulum as accurately as possible

Answer 19

• Use a fiducial mark at the centre of oscillations
• Work out 10 oscillations and then divide by 10 to get the mean

Question 20

Where should you place the fiducial mark

Answer 20

The mark should be at the equilibrium position since this is where the mass moves with greatest speed

Question 21

Describe the Time Period Equation for a Mass-Spring Simple Harmonic System

Answer 21

T = 2π √ (m / k)

Time Period = 2π √ (Mass / Spring Constant)

Question 22

Describe the Energy Transfers that occur in a Mass-Spring Simple Harmonic System

Answer 22

• There are two types of mass-spring system, where the spring is vertical or horizontal
• For the vertical system, kinetic energy is converted to both elastic and gravitational potential energy
• In the horizontal system, kinetic energy is converted only to elastic potential energy

Question 23

Describe the Energy Transfers that occur in any Simple Harmonic System

Answer 23

• Kinetic energy is transferred to potential energy and back as the system oscillates (the type of potential energy depends on the system)
• At the amplitude = maximum amount of potential energy
• As it moves towards the equilibrium position, this potential energy is converted to kinetic energy
• At the centre = maximum kinetic energy
• Then as the system moves away from the equilibrium again, the kinetic energy is transferred to potential energy until it is at a maximum again
• The total energy of the system remains constant (when air resistance is negligible, otherwise energy is lost as heat)

Question 24

Describe the graph of potential enegery against kinetic energy for a simple harmonic system

Answer 24

Question 25

What is Damping

Answer 25

Where the energy in an oscillating is lost to the environment, leading to reduced amplitude of oscillations

Question 26

Describe a Graph of the Effect of Damping

Answer 26

Question 27

Describe the three main types of damping

Answer 27

• Light damping - This is also known as under-damping and this is where the amplitude gradually decreases by a small amount each oscillation
• Critical damping - This reduces the amplitude to zero in the shortest possible time (without oscillating).
• Heavy damping - This is also known as over-damping, and this is where the amplitude reduces slower than with critical damping, but also without any additional oscillations.

Question 28

What are some of the charactertics of a Damping Force

Answer 28

• The size of the force is directly proportional to the negative velocity
• F ∝ - v
• It causes the kinetic energy to be transferred into other forms of energy (such as heat)

Question 29

When do free vibrations/oscillations occur

Answer 29

When no external force is continuously acting on the system, therefore the system will oscillate at its natural frequency

Question 30

When do forced vibrations/oscillators occur

Answer 30

Forced vibrations are where a system experiences an external driving force which causes it to oscillate

Question 31

What happens when the driving force from a forced oscillator is less than, equal to or higher than the natural frequency

Answer 31

• If the driving frequency is less than the natural frequency it causes a small amplitude
• If the driving frequency is higher than the natural frequency it causes a small amplitude
• If the driving frequency is equal to the natural frequency then it causes a very large amplitude (resonance)

Question 32

What is Resonance

Answer 32

Resonance occurs when the driving force is equal to the natural frequency so the system experiences max amplitude oscillations due to an increased amount of energy

Question 33

What are some examples of resonance

Answer 33

• Instruments - an instrument such as a flute has a long tube in which air resonates, causing a stationary sound wave to be formed
• Radio - these are tuned so that their electric circuit resonates at the same frequency as the desired broadcast frequency
• Swing - if someone pushes you on a swing they are providing a driving frequency, which can cause resonance if it’s equal to the resonant frequency and cause you to swing higher
• Bridge - resonance is a negative effect for a bridge for example when the people crossing a bridge they may provide a driving frequency close to the natural frequency so it will begin to oscillate violently which could be very dangerous and damage the bridge

Question 34

How can the oscillations of a bridge be reduced in order to stop resonace from occuring

Answer 34

• Stiffen the structure (by reinforcement)
• Install dampers or shock absorbers

Question 35

What is meant by the term forced oscillation

Answer 35

When the oscillations are caused by a driving force that is periodic to the free oscillations