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## Physics library

### Course: Physics library>Unit 8

Lesson 2: Simple harmonic motion (with calculus)

# Harmonic motion part 3 (no calculus)

Figuring out the period, frequency, and amplitude of the harmonic motion of a mass attached to a spring. Created by Sal Khan.

## Want to join the conversation?

• Sal talks about the simple harmonic motion of a spring and how its formulas are derived. Can anyone please do that for a pendulum? thanks.
• Set the parallel component of the force of gravity as the source of the torque on the pendulum.
τ = r x F = r*mg*sin(Θ) = Iα = mr²α = mr²*d²(Θ)/dt²
where m is the mass of the pendulum and r is the length of the string on the pendulum.
Use a small angle approximation to let sin(Θ) ~= Θ to make the differential equation linear and solvable.
gΘ = r*d²(Θ)/dt²
This equation is now in the same form as the mass spring equation of motion
kx = m*d²(x)/dt²
So solving the 2nd order differential equation you get
Θ(t) = Θi*cos(√(g/r)*t)
where Θi is the initial angle of the pendulum at t=0.
• is the angular velocity a constant value.
if it is a constant value then why is angular acceleration present. because acceleration is just change in velocity and if velocity is constant then there should be no acceleration.
pls help
• You are confusing angular motion with linear. Acceleration, a, is the time rate of change of velocity in in some straight line direction. The spring for example accelerates the mass along a line. Angular frequency, omega, is the number of radians per second (thus the angular) which is just 2*pi*f. The frequency, f, is the number of full cycles per second. Angular frequency is used because it works best with trig functions.

Angular acceleration is not part of SHM.
• does harmonic motion means oscilatting motion?
• From what I've learned, simple harmonic motion has these characteristics:

1. Force is directly proportional to the displacement (`F = -kx` in a spring, for example)
2. This causes the motion to be oscillatory.
3. It can thus be described by the formula `x(t) = A * cos(wt - φ)`
• What if gravity's effect is taken into account? Will the motion of the spring still be considered simple harmonic?
• Yes, it will, because gravity has the same effect at the top as it does at the bottom, so it sort of cancels out.
• Sal and David (see previous section video on equation for...) give different equations for the SHM. They are similiar, but what are the differences? Is there places where I should only use one or is one more general?
David's Equation: x(t)=A sin (2pi/T t)
Sal's Equation: x(t)=A cos sqr root(k/m) *t (see above)
• Sal's is for a mass/spring system.
David's is for any SHM system. By comparison you can tell what the period of Sal's mass spring system must be.
Also in sal's system he is setting the time = 0 when his mass is at a peak. David's equation assumes t = 0 when the system is at equilibrium. That's why one is sin and the other cos.
• I learned that the equation for a harmonic series of waves in a pipe was f=n(v/2L) where f is the frequency, n is the harmonic number, v is the velocity, and L is the length. Where did the equation t=2pi times square root of m divided by k come from?
• According to the equation T=2pi(l/g)^1/2 (time period of a simple pendulum, time period is inversely proportional to the square of gravity and independant of mass, But according to the equation T=2pi(m/k)^1/2, time period is independent of gravity ans directly proportional to the square of mass. What are we supposed to consider?
• Are you comparing a pendulum to a mass/spring system? They're similar, but not the same.
• I have two questions actually:
1. What exactly is omega in this particular equation? I am used to it being angular velocity but because it does not seem like there is angular velocity in this I am a bit confused.
2. Is the t in the equation x(t) = Acos(SQRT(k/m)*t) the period? If no or not necessarily then could you explain how to get t without it being given to you?
(1 vote)
• Omega is still angular velocity. Harmonic motion corresponds to circular motion. Each full cycle is once around a circle.

the t is the time at which you are trying to find the position, x(t)
You could be given a t and then asked to find x. Or you might know x and want to figure out the time at which the system was at point x.
• how is it root of k/m ? wasn't it omega ?
• w = sqrt ( k / m ) or sq. w = ( k / m ) .
• at my beloved Mr.Khan said if there is g the situation will be little bit different but from equation T = 2pi sqrt(m/k) , must not g have any effect on this case?
• Good observation :-). He said that because the figure that he made wouldn't be as simple as it is if we were dealing with gravity. There would be the weight of the object acting downwards (which would affect its position) and a whole lot other things. But eventually the formula would turn out to be the same.(T=2pi sqrt(m/k)). So I think he said that so that we don't get confused by the diagram. That's it. :-)