Further mechanics Flashcards
What is simple harmonic motion
Where an object oscillates either side of an equilibrium point
What is the condition for SHM to take place
The acceleration is directly proportional to the displacement from the equilibrium and acts towards the equilibrium
Why does the acceleration act opposite to the displacement
The displacement increases away from the equilibrium and acceleration is due to the restoring force which brings the system back to equilibrium
How does energy change during SHM
- Total energy (Ek + Ep) remains constant
- As displacement gets further from equilibrium, Ep (which is either elastic potential for a spring or GPE for pendulum increases) increases and Ek decreases
(and vice versa)
Explain how you can draw a graph of displacement against time for an SHM system
- It is a function of Cosine
- Max value = A (symbol for amplitude)
Explain how you can draw a graph of velocity against time for an SHM system
- gradient of displace/time so function of negative sine
- Therefore the phase difference between the v/t graph and x/t graph is π/2
- Max value = wA where w is angular frequency
Explain how you can draw a graph of acceleration against time for an SHM system
- Gradient of velocity/time so function of negative cosine
- Therefore π out of phase with x/t
- Max value at w^2 A
Define frequency in an SHM system
Number of complete cycles per second
(where 1 cycle is e.g. left to right and back to left)
Define time period in SHM system
Time taken for one complete cycle
(where 1 cycle is e.g. left to right and back to left)
Define angular frequency in SHM
Angular displacement per second
w = 2π/T or w = 2πf
What is the relationship between amplitude and frequency (or time period) in SHM
They are independent from each other so f and T remain the same even if oscillations get smaller
Explain the formula relating acceleration and displacement
a = - w^2 x
Because a is directly proportional to x and acts opposite so negative and constant is w^2
(Therefore e.g. the force restoring back to equilibrium doubles as displacement doubles)
Explain the formula for max acceleration
a max = -w^2 A
Because a is proportional to x so max a is when x = A
Explain the formula for max velocity
v = w * sqrt(A^2 - x^2)
and v max is where x = 0
v max = wA
What is the formula for displacement as a function of cosine
Acos(wt)
max displacement is when cos(wt) = 1 so t = 0
How can you find w in an experiment
plot a straight line graph of a against x and w = sqrt(gradient)
Describe the experiment to find the formula for T in a mass spring system
- Suspend a mass and find the time taken for 10 oscillations then divide by 10 to get T
- Repeat for different values of mass, spring constant (by using different combos of series and parallel) and amplitude with separate experiments for each
- You would find that T ∝ sqrt(m) and T ∝ sqrt(1/k) and A has no affect
So T = 2π*sqrt(m/k)
Also to remember: use a fiducial marker as a point of reference for midpoint and wear safety goggles
What are free vibrations
Occur when no external forces act on the system so oscillates at natural frequency (known as resonant frequency)
Imagine swing if you let it go but don’t push it
What are forced vibrations
Occur when you apply an external driving force to make the system oscillate
Imagine pushing a swing
Explain resonance
Where the amplitude of oscillations drastically increase because the driving force was applied equal to the resonant frequency
Give 3 examples of resonance
- Flutes which cause a column of air to resonate creating a stationary wave
- Radios which are tuned so that the circuit resonates at the same frequency as the broadcast for max signal
- A swing where someone pushes you at the resonant frequency causing you to swing higher
What is damping
Where energy is lost to surroundings due to frictional forces like air resistance (known as damping forces)
What is a use of damping
To reduce the number of oscillations which stops resonance from occurring which could be unsafe i.e.
Describe light damping
Where the amplitude decreases slowly over time
Describe heavy damping
Where the amplitude decreases over time but more quickly than with light damping
Describe critical damping
Amplitude goes to 0 straight away (think of a cos graph from 0 to 90 and then straight line at 0 for the rest of the time)
Describe overdamping
There are no oscillations like with critical damping but it takes a lot longer to get to 0
(the cos graph from 0 to 90 stretched the the x axis)
Explain how damping affects the amplitude when resonance occurs
- If you plot a graph of amplitude against driving frequency, the graph would look like a normal distribution curve with some A when DF is 0 and no A at high driving frequencies and a peak A when the DF = RF
- Lighter damping causes the peak amplitude to be higher and sharper
- sharper means less range of driving frequencies cause resonance
Describe the condition for circular motion
Where the (centripetal) force applied is always perpendicular to velocity
Define angular speed
The angle that the object rotates through per second
What is the relationship between angular speed and linear speed
w = v/r
Define frequency and time period for circular motion
- Frequency is the number of revolutions per second
- Time period is time taken for 1 full revolution
Explain the relationship between angular speed and frequency for circular motion
- w = 2π/T (think of 1 full cycle)
- So w = 2πf
Explain acceleration in circular motion
- The direction is constantly changing even though speed is constant - - Changing velocity causes centripetal acceleration which acts towards centre of circle
- a = w^2 r = v^2 / r
Explain centripetal force in circular motion
- Centripetal acceleration which acts towards centre must be caused by centripetal force which also acts towards centre
- F = mv^2/r = mw^2 r
Explain what would happen to the roller coaster on a loop if the track ran out
- When travelling on the loop centripetal force acts towards centre and velocity is perpendicular so at tangents to circle
- If track runs out, the carriage would fly off in the direction of velocity so at a tangent to circle
Example: swinging bucket of water around in a circle with a string
Explain the forces acting on the bucket of water at different points in the motion
- There is a support force in the string bringing it to centre of circle and weight acting down of bucket which together make up centripetal force
- At the top of circle centripetal force = S +Mg
- At the bottom CF = S-Mg
- At the right or left CF = S
Example: swinging bucket of water around in a circle with a string
Explain why the water doesn’t fall out of the bucket
- Water is in circular motion so the forces cause its velocity to be at a tangent to a circle and not straight down
Example: swinging bucket of water around in a circle with a string
Explain the conditions required to maintain the bucket of water in circular motion without water falling out
- At the top mv^2 /r = s + mg
- Only at the top, weight acts down and if it is greater than the centripetal force, the water will fall out
- Therefore, to maintain circular motion mv^2 /r ≥ mg so s ≥ 0
- v^2 /r ≥ g meaning you should increase v or decrease r to stop water falling out
- When the water falls out mv^2 /r < mg which would imply s<0 i.e. impossible so no circular motion
