S.H.M. Mathematics Flashcards
for object moving in a circle, what is the equation for its angular velocity
- w = 0 / t
- 0 = theta
what is the basic equation for period
T = 1/f
what is the equation for period that includes the angular velocity of the object
T = 2pi / w
therefore what is another equation for angular velocity including f
- w = 2pi / T
- but as f = 1 / T
- w = 2pi f
does an object need to be moving in a circle in order to calculate its angular velocity and why
- no
- because its motion can be the projection of motion in a circle
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
- 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
given the equation w = 0 / t, what is another way for writing the horizontal displacement of the object from the centre
- 0 = wt
- so x = r cos(wt)
if you were trying to calculate the displacement of an object on a pendulum swing, how would you modify that equation
- the radius r would become the amplitude of the pendulum, A
- giving x = A cos(wt)
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
- you know T = 2s
- so the only equation you can use is w = 2pi / T
- 2pi / 2 = 3.14
where will the pendulum be after 174.5 seconds
- 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
what equation do you get when combining F = -kx and x = A cos(wt)
F = -k(A cos(wt))
therefore how would you derive the equation a = -k(A cos(wt)) / m
- F = ma
- a = F / m
- as F = -k(A cos(wt))
- a = -k(A cos(wt)) / m
what does this equation show us about the relationship between acceleration and displacement
- 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
what does that relationship already tell you about the relationship between the displacement-time and acceleration-time graphs
- the displacement-time graph would simply be flipped to make the acceleration-time graph
- they would be mirrors of each other
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
- differentiate it
- dx/dt = cos0
- then plug in the value for 0 you need
- because 0 itself is a function of time
is x = A cos(wt), what would be the function for the velocity of the oscillating object
- differentiate to get velocity function with respects to time
- dx/dt = v = -Aw sin(wt)
what would be the function for acceleration
- differentiate the velocity function with respects to time
- d^2x/dt^2 = a = -Aw^2 cos(wt)
what could the acceleration function be further simplified to and why
- a = -xw^2
- because x = A cos (wt)
what is the equation for working out the restoring force constant k, in terms of angular velocity and mass
k = mw^2
what is the relationship between the velocity-time graph and the acceleration-time graph
- 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
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)
- flip the d-time graph over to get the the a-time graph
- move the line pi / 2 radians to the right
what is the equation for working out the max velocity if an oscillating object
v = 2pi f x
What is x in that equation
Maximum displacement from the equilibrium (amplitude)
