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## 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 says that the expressions of time period and frequency are independent of the displacement but it is clear that the two expressions are dependent on 'k' which is the spring constant. My question is that isn't 'k' dependent on the displacement 'A' if we go back to the equation F= -kx, if so then the expressions for time period and frequency are indirectly dependent on the displacement. I hope this makes sense and is my reasoning right? • 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.
• What if gravity's effect is taken into account? Will the motion of the spring still be considered simple harmonic? • does harmonic motion means oscilatting motion? • 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?   