Further Mechanics

Question 1

What is the definition of radians?

Answer 1

The angle in radians is equal to the arc-length divided by the radius of the circle.

Question 2

How do you convert between radians and degrees?

Answer 2

angle in radians = angle in degrees x π/180

Question 3

What is angular speed? What is it’s equation?

Answer 3

The angular speed is the angle an object rotates through per second.
ω = ϴ/t

Question 4

How do you write the angular speed formula in terms of the linear speed?

Answer 4

ω = v/r

Question 5

What is the definition of frequency? What is the definition of period?

Answer 5

The frequency, f, is the number of complete revolutions per second (revs^-1 or hetz, Hz). The period, T, is the time taken for a complete revolution (in seconds).

Question 6

How are freuqency and period linked? How can you write the angular speed formula in terms of frequency or period?

Answer 6

f = 1/T
For a complete circle, an object turns through 2π radians in a time T, So: ω = 2π/T
T = 1/f, so: ω = 2πf

Question 7

What is centripetal acceleration?

Answer 7

Centripetal acceleration is the rate of change of velocity when something is travelling in a circle (velocity is changing due to the direction constantly changing, not speed). Always directed towards the centre of the circle.

Question 8

What is centripetal force? How do you get the equations for centripetal force?

Answer 8

The force which causes the centripetal acceleration. Always acts towards the centre of the circle. To get the equations, use F = ma and substitute in the equations for centripetal acceleration in place of a.

Question 9

What is simple harmonic motion?

Answer 9

An object moving with simple harmonic motion oscillates to and fro, either side of the equilibrium postion.

Question 10

What is the definition of simple harmonic motion?

Answer 10

An oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position, and is directed towards the equilibrium. a ∝ -x where x is the displacement.

Question 11

Describe the graphs of displacement, velocity and acceleration against time.

Answer 11

• Displacement-time graph varies as a cosine or sine wave with a maximum value A (the amplitude).
• Velocity-time graph is the gradient of the displacement-time graph. Maximum value of ωA.
• Acceleration-time graph is the gradient of the velocity-time graph. Maximum value of ω^2 A

Question 12

What is phase difference?

Answer 12

Phase difference is a measure of how much one wave lags behind another wave, and can be measure in degrees, radians, or fractions of a cycle.

Question 13

How does an object in SHM exchange potential and kinetic energies? What happens to the energies at equilibrium position and at amplitude?

Answer 13

-As the object moves towards equilibrium, the restoring force does work on the object so transfers so Ep to Ek.
-When object is moving away from equilibrium, Ek is transfered back to Ep.
-At equilibrium, Ek is maximum and Ep is zero.
At amplitude, Ek is zero and Ep is a maximum.

Question 14

What is the sum of the potential and kinetic energy called?

Answer 14

The sum of Ep and Ek is called the mechanical energy.

Question 15

What is a useful way of understanding the maths behind SHM?

Answer 15

Think about it as the ‘projection’ of circular motion onto a horizontal plane (ball spinning in a circle - from above will look like a circle but from side will look like it’s oscillating from side to side).

Question 16

What is angular frequency?

Answer 16

Angular frequency is the equivalent ‘component’ to ω in circular motion.

Question 17

For a mass on a spring, how is the size and direction of the restoring force worked out?

Answer 17

Hooke’s law - F = kΔL

Question 18

How would you set up an experiment to investigate the mass-spring system?

Answer 18

• Attach a trolley to a spring, pull it to one side by a certain amount and then let go.
• Trolley will oscillate back and forth
• Can measure the period T by using a computer to plot the displacement-time graph from a data logger connected to a position sensor

Question 19

Name 3 things you can change in the mass-spring system to change the results.

Answer 19

You can change the mass on the spring, the spring constant and the amplitude.

Question 20

What will happen if you change the mass on the mass-spring system?

Answer 20

Since T ∝ √m, the square of the period, T^2, should be proportional to the mass.

Question 21

What will happen if you change the spring constant on the mass-spring system?

Answer 21

Since T ∝ √1/k, the square of the period, T^2, should be proportional to the inverse of the spring constant.

Question 22

What will happen if you change the amplitude on the mass-spring system?

Answer 22

Since T doesn’t depend on amplitude, A, there should be no change in the period.

Question 23

Name 2 other types of simple harmonic oscillations (other than mass-spring system).

Answer 23

The simple pendulum and a U-tube containing some water.

Question 24

How can you investigate the formula for the period of a simple pendulum system? What will happen if you change the variables on this experiment?

Answer 24

You can use a simple pendulum attached to an angle sensor and computer.

• Use the computer to plot a displacement-time graph and read off the period, T, from it. Make sure you calculate the average period over several oscillations to reduce the percentage error.
• As for a spring, you can change one variable at a time and measure what happens
• Since the period , T, is proportional to the square root of the length of the pendulum, varying l should show that T^2 ∝ l
• T is independent of the mass of the bob, m, and the amplitude of the oscillation, A, so varying these will not change T.

Question 25

What is a free vibration?

Answer 25

Free vibrations involve no transfer of energy to or from the surroundings - will keep oscillating with the same amplitude forever.

Question 26

What is a forced vibration?

Answer 26

Forced vibrations happen when there’s an external driving force. Frequency of this force is the driving frequency.

Question 27

What is resonance? What is the phase difference between the driver and the oscillator when the system is resonating?

Answer 27

When the driving frequency approaches the natural frequency, the system gains more and more energy and so vibrates with rapidly increasing amplitude. At resonance phase difference between the driver and the oscillator is 90 degrees.

Question 28

Give 4 examples of resonance in the real world.

Answer 28

• A radio (tuned so electric circuit resonates at same frequency as radio station)
• A glass (resonates when driven by a sound wave at right frequency)
• Organ pipe (column of air driven by the motion of air at base - creates a stationary wave in pipe)
• A swing (resonates if it’s driven by someone pushing i at it’s natural frequency)

Question 29

What is damping? What is an example of damping in the real world?

Answer 29

Damping is when an oscillating system loses energy to it’s surroundings (through frictional forces like air resistance). Example is shock absorbers in a car’s suspension - squash oil through a hole to provide damping force.

Question 30

What is light and heavy damping?

Answer 30

Lightly damped systems take a long time to stop oscillating, and their amplitude only reduces a small amount each period. Heavily damped systems take less time to stop oscillating.

Question 31

What is critical damping? Give an example.

Answer 31

Critical damping reduces the amplitude in the shortest possible time. (e.g. car suspension systems)

Question 32

What is overdamping? Give an example.

Answer 32

Overdamped systems take longer to return to equilibrium than a critically damped system. (e.g. heavy doors, so they don’t slam shut)

Question 33

What happens to the phase difference when the driving force is less/more than the natural frequency?

Answer 33

When the driving force is less, the two are in phase. When the driving force is much greater, the two will oscillate in antiphase (180 degrees)