Further Mechanics Flashcards
Describe motion in a circular path at a constant speed
Motion in a circular path at constant speed implies there is an acceleration and requires a centripetal force
Which direction does a Centripetal Force act
A centripetal force always acts towards the centre of the circle
What is Angular Speed
The angle an object moves through per unit time
What are the Equations for Angular Speed
ω = v / r
Angular Speed = Velocity / Distance
ω= 2πf
Angular Speed = 2π x Frequecny
What are the Equations for Centripetal Acceleration
a = v2 / r = ω2r
Acceleration = Velocity2 / Distance = Angular Speed2 X Distance
What are the Equations for Centripetal Force
F = mv2 / r = mω2r
Centripetal Force = Mass x Velocity2 / Distance = Mass x Angular Speed2 x Distance
When is an object experencing simple harmonic motion
When its acceleration is directly proportional to displacement from the equilibrium point and is in the opposite direction
What is the equation for accleration in simple harmonic motion
a = -ω2x
Accleration = - Angular Speed2 x Displacement
What is the equation for displacement in simple harmonic motion
x = A Cos(ωt)
Displacement = Amplitude x Cos(Angular Speed x Time)
What is the equation for velocity in simple harmonic motion
v = ± ± ω √(A2 - x2)
Velocity = ± Angular Speed √(Amplitude2 - Displacement2)
When are velocity and acceleration at their maximum and minimum on a pendulm
- Velocity Max = Centre
- Velcoity Zero = Amplitude
- Acceleration Max = Amplitude
- Accleration Zero = Centre
What are the equations for velocity max and acceleration max
- Vmax = ωA
- amax = ω2A
Describe the displacement-time graph of simple harmonic motion
Describe the Velocity-Time Graph of Simple Harmonic Motion
Describe the Acceleration-Time Graph of Simple Harmonic Motion
Describe the Time Period Equation for a Simple Harmonic Pendulum
T = 2π √(L / g)
Time Period = 2π √ (Length of the String / Acceleration due to Gravity)
Explain why the simple harmonic pendulum only works for small angles
- 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of the above formula a small angle approximation is used, and so for larger initial angles this approximation is no longer valid, and would not be a good model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
What are the Limitations of the Simple Harmonic Motion Equations for a Simple Pendulum
- The amplitude is small
- The string is inextensible
- The bob used is a point mass
How would you measure the time period of the oscillations of a simple pendulum as accurately as possible
- Use a fiducial mark at the centre of oscillations
- Work out 10 oscillations and then divide by 10 to get the mean
Where should you place the fiducial mark
The mark should be at the equilibrium position since this is where the mass moves with greatest speed
Describe the Time Period Equation for a Mass-Spring Simple Harmonic System
T = 2π √ (m / k)
Time Period = 2π √ (Mass / Spring Constant)
Describe the Energy Transfers that occur in a Mass-Spring Simple Harmonic System
- 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
Describe the Energy Transfers that occur in any Simple Harmonic System
- 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)
Describe the graph of potential enegery against kinetic energy for a simple harmonic system
What is Damping
Where the energy in an oscillating is lost to the environment, leading to reduced amplitude of oscillations
Describe a Graph of the Effect of Damping
Describe the three main types of damping
- 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.
What are some of the charactertics of a Damping Force
- 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)
When do free vibrations/oscillations occur
When no external force is continuously acting on the system, therefore the system will oscillate at its natural frequency
When do forced vibrations/oscillators occur
Forced vibrations are where a system experiences an external driving force which causes it to oscillate
What happens when the driving force from a forced oscillator is less than, equal to or higher than the natural frequency
- 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)
What is Resonance
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
What are some examples of resonance
- 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
How can the oscillations of a bridge be reduced in order to stop resonace from occuring
- Stiffen the structure (by reinforcement)
- Install dampers or shock absorbers
What is meant by the term forced oscillation
When the oscillations are caused by a driving force that is periodic to the free oscillations
