Further mechanics

Question 1

What is simple harmonic motion

Answer 1

Where an object oscillates either side of an equilibrium point

Question 2

What is the condition for SHM to take place

Answer 2

The acceleration is directly proportional to the displacement from the equilibrium and acts towards the equilibrium

Question 3

Why does the acceleration act opposite to the displacement

Answer 3

The displacement increases away from the equilibrium and acceleration is due to the restoring force which brings the system back to equilibrium

Question 4

How does energy change during SHM

Answer 4

• 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)

Question 5

Explain how you can draw a graph of displacement against time for an SHM system

Answer 5

• It is a function of Cosine
• Max value = A (symbol for amplitude)

Question 6

Explain how you can draw a graph of velocity against time for an SHM system

Answer 6

• 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

Question 7

Explain how you can draw a graph of acceleration against time for an SHM system

Answer 7

• Gradient of velocity/time so function of negative cosine
• Therefore π out of phase with x/t
• Max value at w^2 A

Question 8

Define frequency in an SHM system

Answer 8

Number of complete cycles per second

(where 1 cycle is e.g. left to right and back to left)

Question 9

Define time period in SHM system

Answer 9

Time taken for one complete cycle

(where 1 cycle is e.g. left to right and back to left)

Question 10

Define angular frequency in SHM

Answer 10

Angular displacement per second

w = 2π/T or w = 2πf

Question 11

What is the relationship between amplitude and frequency (or time period) in SHM

Answer 11

They are independent from each other so f and T remain the same even if oscillations get smaller

Question 12

Explain the formula relating acceleration and displacement

Answer 12

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)

Question 13

Explain the formula for max acceleration

Answer 13

a max = -w^2 A

Because a is proportional to x so max a is when x = A

Question 14

Explain the formula for max velocity

Answer 14

v = w * sqrt(A^2 - x^2)
and v max is where x = 0
v max = wA

Question 15

What is the formula for displacement as a function of cosine

Answer 15

Acos(wt)

max displacement is when cos(wt) = 1 so t = 0

Question 16

How can you find w in an experiment

Answer 16

plot a straight line graph of a against x and w = sqrt(gradient)

Question 17

Describe the experiment to find the formula for T in a mass spring system

Answer 17

1. Suspend a mass and find the time taken for 10 oscillations then divide by 10 to get T
2. Repeat for different values of mass, spring constant (by using different combos of series and parallel) and amplitude with separate experiments for each
3. 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

Question 18

What are free vibrations

Answer 18

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

Question 19

What are forced vibrations

Answer 19

Occur when you apply an external driving force to make the system oscillate

Imagine pushing a swing

Question 20

Explain resonance

Answer 20

Where the amplitude of oscillations drastically increase because the driving force was applied equal to the resonant frequency

Question 21

Give 3 examples of resonance

Answer 21

1. Flutes which cause a column of air to resonate creating a stationary wave
2. Radios which are tuned so that the circuit resonates at the same frequency as the broadcast for max signal
3. A swing where someone pushes you at the resonant frequency causing you to swing higher

Question 22

What is damping

Answer 22

Where energy is lost to surroundings due to frictional forces like air resistance (known as damping forces)

Question 23

What is a use of damping

Answer 23

To reduce the number of oscillations which stops resonance from occurring which could be unsafe i.e.

Question 24

Describe light damping

Answer 24

Where the amplitude decreases slowly over time

Question 25

Describe heavy damping

Answer 25

Where the amplitude decreases over time but more quickly than with light damping

Question 26

Describe critical damping

Answer 26

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)

Question 27

Describe overdamping

Answer 27

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)

Question 28

Explain how damping affects the amplitude when resonance occurs

Answer 28

- 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

Question 29

Describe the condition for circular motion

Answer 29

Where the (centripetal) force applied is always perpendicular to velocity

Question 30

Define angular speed

Answer 30

The angle that the object rotates through per second

Question 31

What is the relationship between angular speed and linear speed

Answer 31

w = v/r

Question 32

Define frequency and time period for circular motion

Answer 32

- Frequency is the number of revolutions per second - Time period is time taken for 1 full revolution

Question 33

Explain the relationship between angular speed and frequency for circular motion

Answer 33

- w = 2π/T (think of 1 full cycle) - So w = 2πf

Question 34

Explain acceleration in circular motion

Answer 34

- 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

Question 35

Explain centripetal force in circular motion

Answer 35

- Centripetal acceleration which acts towards centre must be caused by centripetal force which also acts towards centre - F = mv^2/r = mw^2 r

Question 36

Explain what would happen to the roller coaster on a loop if the track ran out

Answer 36

- 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

Question 37

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

Answer 37

- 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

Question 38

Example: swinging bucket of water around in a circle with a string Explain why the water doesn't fall out of the bucket

Answer 38

- Water is in circular motion so the forces cause its velocity to be at a tangent to a circle and not straight down

Question 39

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

Answer 39

- 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