S.H.M. Mathematics

Question 1

for object moving in a circle, what is the equation for its angular velocity

Answer 1

• w = 0 / t

- 0 = theta

Question 2

what is the basic equation for period

Answer 2

T = 1/f

Question 3

what is the equation for period that includes the angular velocity of the object

Answer 3

T = 2pi / w

Question 4

therefore what is another equation for angular velocity including f

Answer 4

• w = 2pi / T
• but as f = 1 / T
• w = 2pi f

Question 5

does an object need to be moving in a circle in order to calculate its angular velocity and why

Answer 5

• no

- because its motion can be the projection of motion in a circle

Question 6

theres a circle with an object, connected to the centre, that can move along the circumference of the circle. the real positive axis of the circle (given that its split into 4 quadrants like youre used to) is point A. however the object moves along the circumference to point B (sill in Q1). the angle between point A and B is theta (0) and the length of the string connecting the object to the centre is r. how would you derive an equation to calculate the horizontal distance (x) between the centre and point B

Answer 6

• r would be the hypotenuse
• so the angle between x and r is theta (0)
• this means we can use cos to calculate x
• cos0 = x / r
• so x = r cos0

Question 7

given the equation w = 0 / t, what is another way for writing the horizontal displacement of the object from the centre

Answer 7

• 0 = wt

- so x = r cos(wt)

Question 8

if you were trying to calculate the displacement of an object on a pendulum swing, how would you modify that equation

Answer 8

• the radius r would become the amplitude of the pendulum, A

- giving x = A cos(wt)

Question 9

a pendulum is released from its amplitude position that is 10cm from the middle. it swings completely through one cycle every two seconds. what is its angular velocity

Answer 9

• you know T = 2s
• so the only equation you can use is w = 2pi / T
• 2pi / 2 = 3.14

Question 10

where will the pendulum be after 174.5 seconds

Answer 10

• you would use the displacement equation for pendulums for this
• x = A cos(wt)
• A = 10cm = 0.1m
• x = 0.1 cos(3.14 * 174.5)
• x = 0.027m

Question 11

what equation do you get when combining F = -kx and x = A cos(wt)

Answer 11

F = -k(A cos(wt))

Question 12

therefore how would you derive the equation a = -k(A cos(wt)) / m

Answer 12

• F = ma
• a = F / m
• as F = -k(A cos(wt))
• a = -k(A cos(wt)) / m

Question 13

what does this equation show us about the relationship between acceleration and displacement

Answer 13

• acceleration acts in the opposite direction to displacement
• when displacement is 0 so is acceleration
• and when displacement is at its max, acceleration will be too

Question 14

what does that relationship already tell you about the relationship between the displacement-time and acceleration-time graphs

Answer 14

• the displacement-time graph would simply be flipped to make the acceleration-time graph
• they would be mirrors of each other

Question 15

for the displacement-time graph, its line can be modeled as the function x = sin0. what is the easiest way to calculate the velocity of an oscillating object at any given time using this equation

Answer 15

• differentiate it
• dx/dt = cos0
• then plug in the value for 0 you need
• because 0 itself is a function of time

Question 16

is x = A cos(wt), what would be the function for the velocity of the oscillating object

Answer 16

• differentiate to get velocity function with respects to time
• dx/dt = v = -Aw sin(wt)

Question 17

what would be the function for acceleration

Answer 17

• differentiate the velocity function with respects to time

- d^2x/dt^2 = a = -Aw^2 cos(wt)

Question 18

what could the acceleration function be further simplified to and why

Answer 18

• a = -xw^2

- because x = A cos (wt)

Question 19

what is the equation for working out the restoring force constant k, in terms of angular velocity and mass

Answer 19

k = mw^2

Question 20

what is the relationship between the velocity-time graph and the acceleration-time graph

Answer 20

• the velocity-time graph is quarter of a wavelength ahead of the acceleration-time graph
• so its just a translation of the a-time graph of pi / 2 to the right

Question 21

therefore if you were given a displacement-time graph, how would you draw the velocity-time graph (other than differentiating its equation then just plotting lol)

Answer 21

• flip the d-time graph over to get the the a-time graph

- move the line pi / 2 radians to the right

Question 22

what is the equation for working out the max velocity if an oscillating object

Answer 22

v = 2pi f x

Question 23

What is x in that equation

Answer 23

Maximum displacement from the equilibrium (amplitude)