3. What Goes Around Comes Around

Question 1

What happens to a person at the top of a pendulum theme park ride such as a ‘swinging pirate ship’ and why?

Answer 1

The rider loses contact with the seat and experiences the sensation of weightlessness as there is no reaction force from the seat

Question 2

Give the definitions for

• Amplitude
• Time period
• Frequence

Answer 2

• Amplitude is the maximum displacement of an object from the equilibrium position
• Time period is the time taken for one complete cycle of the oscillation
• Frequency is the number of cycles per second of the object

Question 3

What is the shape of a displacement time graph for a pendulum? What does the gradient of the graph represent?

Answer 3

A cosine curve

The gradient represents the velocity

Question 4

What is the shape of a velocity time graph for a pendulum? What does the gradient represent?

Answer 4

A negative sine curve

The gradient represents the acceleration

Question 5

What is the shape of an acceleration time graph for a pendulum?

Answer 5

A negative cosine curve

Question 6

Define simple harmonic motion.

Answer 6

Oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position

Question 7

What are the two conditions of simple harmonic motion? What equation can this be represented by (and what do each of the terms stand for)?

Answer 7

• Acceleration is proportional to the displacement from equilibrium
• Acceleration is in the opposite direction to the displacement

a= -kx
a is acceleration
k is a constant which depends upon the time period (equal to (2πf)^2
x is the displacement from equilibrium

Question 8

What is the equation for the SHM curve of displacement against time?

Answer 8

x= Acos(2πft)

Question 9

Since it is not expected of you to solve the simple harmonic equation mathemetically, it is necessary to make sure that the equation is a sensible solution.

Give examples to why x=Acos(2πft) is a sensible solution (aka how do you show it is correct/sensible) (hard question sorry lol) Make reference to the displacement time graph that represents simple harmonic motion.

Answer 9

• Shape of the displacement time graph is a cosine, so the shape of the equation and the curve match
• Max value of sine or cosine is 1 so if the cosine term cos(2πft) equals 1 then x=A which is sensible as the amplitude is the maximum displacement
• Since x and A have the same units, the complete cosine term can have no units because 2π have no units and frequency has the units s^-1 (Hz) and time has the units s so they cancel out giving a dimensionless quantity so this part of the equation fits as well
• If one period had elapsed from whenever we started measuring the oscillation and since f=1/T then fT=1 and so x=Acos2π. The cosine of 2π radians is equal to 1 so again the equation makes sense

Question 10

What is the time period of a pendulum dependent on and what is it independent of?

Answer 10

• It is dependent on the length of the pendulum

- It is independent of the mass of the pendulum bob

Question 11

Show that T^2 (time period squared) of a pendulum is proportional to l (the length of the pendulum). What would the graph of T^2 against l look like?

Answer 11

T= 2π √l/g
T^=(4π^2/g) l
Since (4π^2/g) is a constant T^2 is proportional to l.

The graph of period squared against l should give a straight line graph that goes through the origin.

Question 12

Describe how energy changes during the motion of a pendulum.

Answer 12

• When the bob is displaced from equilibrium, it gains gravitational potential energy
• When the pendulum falls and is at the equilibrium position, it has maximum kinetic energy
• The maximum kinetic energy is equal to the gain in gravitational potential energy that occurred when lifting the pendulum (displacing it)
• Assuming there are no energy losses (eg through air resistance) the total energy of the pendulum remains constant

Question 13

In a simple pendulum, when is velocity greatest and when is acceleration greatest and why?

Answer 13

• Velocity is greatest when the pendulum is passing through the equilibrium point because this is when it has maximum kinetic energy (all gravitational potential energy has been converted to kinetic energy)
• Acceleration is greatest when the displacement from equilibrium is greatest because acceleration is proportional to the displacement from equilibrum

Question 14

What is the maximum kinetic energy of a pendulum proportional to?

Answer 14

Proportional to the square of the amplitude

Question 15

How energy changes with time

Answer 15

See textbook

Question 16

Describe how the vertical reaction force varies on a rider in a roller-coaster ride with a sinusoidal track.

Answer 16

In a ‘trough’ the carriage is starting to move upwards whilst the person is moving downwards so here they experience the greatest g force and have the greatest reaction force.

At a ‘crest’ the carriage is starting to move downwards whilst the person is moving upwards so the rider gains the experience of weightlessness as the reaction force is very small