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### Course: Class 11 Physics (India)>Unit 18

Lesson 1: Introduction to simple harmonic motion

# Equation for simple harmonic oscillators

David explains the equation that represents the motion of a simple harmonic oscillator and solves an example problem. Created by David SantoPietro.

## Want to join the conversation?

• Shouldn't the cosine function at 1 point compress and reach zero? I had his query because I thought that the block eventually reaches the equilibrium position.. It doesn't really have this constant oscillation between the maximum value and minimum value. Also, why has he used particularly he cosine function?
• Well, this is the idealized model, where the friction is neglected, so we assume that the block oscillates forever. As you have written it does not REALLY have this constant oscillation, but in this case it is "unreally".
He used cosine function just as an example. He assumed that at the time=0, the block is at the maximum value, but this was just an arbitrary assumption.
• What if the graph doesn't start at a miximum, minimum or 0?
• actually that situation is highly impossible....'cause in that case, the law of conservation of energy would not be valid anymore. just think about it.......if it started at 0.2 meters from the mean position, the amplitude keeps decreasing.....it can never keep increasing just like that......if it was pulled to 0.2 meters then the next time it may stretch upto only 0.18 meters probably. Even the example shown in the video is only hypothetical....no oscillation keeps going on on its own.......it stops at some time. So your given condition is actually never possible, even in a force free field.
• What does this function output again?
• Hello Ammar,

These oscillators can be tricky. We start with a physical movement such as a mass moving up and down on the end of a spring. We finish with trigonometry. At first this is very nonintuitive but I encourage you to keep working. This modeling of oscillators based on trigonometry is a very powerful technique.

Regards,

APD
• Will the spring vibrate infinitely in space or in vacumm
• As long as there is no friction, the vibration will continue indefinitely. In space, there will still be some internal friction in the spring itself as it stretches and compresses.
• Why don't you account for the phase angle? In my textbook it has the equation as x = A cos ( wt + (greek letter phi)). It says Greek letter phi = phase angle. Thank you
• In the video, the spring starts from the amplitude [At t=0]
Therefore we use x(t)=Acos(wt).

If we start from some distance 'x', the equation of motion will be
x(t)=Acos(wt + Φ)
At t=0
x will NOT be A,
It will be x = Acos(Φ).
Hope you understood...
• Sine and Cosine are just shifted versions of each other. Does it really matter which trig function I use ?
(1 vote)
• No, it does not. You just need to adjust the phase angle.
• Can the equation Fs=-Kx be applied to the physics in a pirate ship amusement park ride? Where the ship has simple harmonic motion going back and fourth?
• I think so, though the pirate ship is a more of a pendulum than a spring. The displacement multiplied by k is the force.
(1 vote)
• Can there be more than one function that has the same amplitude and period ?
• You can use either sine or cosine, but you have to adjust the phase so the displacement at t=0 is correct
• At what does he mean by having T alone won't be enough to represent the period? Where did he get cos(0)=1? Overall, what does it mean by a function of time?
• David meant that simply having x = Acos(t) (where x is displacement and t is time) wouldn't be a very god description of every case of simple harmonic oscillation. This is because the period of an oscillator is variable and dependent on the mass and spring constant of the oscillator. Because the period won't be the same for every problem, we have to come up with a more complex definition to use for time.
He gets cos(0) = 1 from trig. If you have an imaginary right triangle (that couldn't exist in real life) with a 0 degree angle, the cosine would be the adjacent over the hypotenuse. Since one angle of this right triangle is 90 degrees and the other is 0 degrees, the third angle must also be 90. Since two angles are the same, the triangle is isosceles and the two 90 degree sides would be the same. If you draw it out in your head, one of the congruent sides is adjacent to the 0 deg angle, and one is the hypotenuse. Thus, when you compute cos(0), you're doing x/x, which is 1.
When it says "as a function of time", the video means that time is the independent variable in this scenario, that's all.

Hope this helps!