Periodic Motion and the Equation of SHM

Transcript

0:00Introduction

Hello everyone! One small puzzle before we start. Take a block on a smooth table, tied to a spring. Pull it one centimetre, let go, and it slides back and forth. Now pull it four centimetres instead. Four times the distance, so each oscillation takes longer, right? Hold on to your answer. Today we learn simple harmonic motion, SHM, and its equation. It comes up often in JEE, so let's begin.

0:29What makes a motion SHM?

So, a periodic motion repeats after a fixed time, the period, capital T, okay? Now picture a particle moving to and fro about a point O, the mean position, as in the question on screen. Let me ask you this: when the particle is to the right of O, which way must its acceleration, small a, point? Towards O, yes, exactly! On the left, it points right, again towards O, see? That is the first condition for SHM about O: the acceleration always points towards the mean position. Right? Second, its size is proportional to x, the displacement from O, so small a equals minus omega squared, times x. Omega, the Greek letter like a rounded w, is a constant, and the little two means squared. The minus sign makes the acceleration opposite to the displacement. Clear? Now multiply by the mass m: the force F equals m times a, which is minus m times omega squared, times x. We call m times omega squared the force constant k, so F equals minus k times x. This is the restoring force, always pulling the particle back. Got it?

1:41The equation of SHM

So, at any time t, x equals capital A, times sine of, omega t plus phi, in brackets; phi is the Greek letter like a circle with a line through it. Or with a cosine, capital A cos of, omega t plus phi prime; the little dash marks a different constant. Okay? Capital A is the amplitude: sine stays between minus one and one, so x stays between minus capital A and plus capital A, on either side of O, right? The whole angle, omega t plus phi, is called the phase, and it changes with time. And phi, its value at t equals zero, is the initial phase, or the epoch. With me?

2:21Period and frequency

Now, from k equals m times omega squared, the angular frequency omega is root k over m; the tick is the root sign, the slash means over. So, by how much must omega t grow for the motion to repeat? By two pi, yes, exactly, one full turn! So the period, capital T, is two pi over omega, where pi is the same pi you know from the circle, that is two pi times root m over k. And the frequency f, oscillations per second, is one over capital T, okay? Now, remember our spring puzzle? Look at this formula for the period. Where is the amplitude in it? It is nowhere, there is no capital A in it at all! The period depends only on m and k. One centimetre or four, each oscillation takes the same time; the block goes farther, but also faster. That was the trap. Got it?

3:18SHM or only periodic?

Now, a useful test. Here, x of t means x as a function of time. It is SHM only if it is one sine or cosine wave of one frequency, perhaps shifted by a constant. Okay? So tell me, sine omega t plus cos omega t has two terms. Is it SHM? Yes, it is SHM, because both terms have the same omega. They combine into one sine, root two times sine of, omega t plus pi over four: one frequency. Got it? But sine omega t plus sine two omega t is different. It repeats, so it is periodic, but it mixes two frequencies, omega and two omega, so it is not simple harmonic. Clear? Now a JEE trap: sine squared omega t, where the little two after sine means squared. It never goes below zero, so it cannot be SHM, can it? Careful! This one really is simple harmonic. Sine squared omega t equals one half, minus one half cos two omega t: one cosine, shifted by one half. So it is SHM about x equals one half, with angular frequency two omega. With me?

4:30The reference circle

Now, a lovely picture: the reference circle. Point P goes round a circle of radius capital A at a steady rate, at the angle omega t plus phi from the x axis. The arrow at P is its velocity, along the circle, of size v equals capital A times omega. Okay? Now drop a perpendicular from P onto the vertical diameter, the dashed line. Its foot is N, the red point. As P goes round, what does N do? It moves up and down along the diameter, in SHM! Its position, y sub N, with the small N below the y, is capital A times sine of, omega t plus phi. Same amplitude, same omega as the circle. Right?

5:13Example 1: period and phase of x = 5 sin(4 pi t + pi/3)

Alright, here is example one. x equals five times sine of, four pi times t, plus pi over three, in centimetres. Find the amplitude, period, frequency, initial phase, and the position at t equals zero. Pause the video and try it first. Done? Okay, let's check. Comparing with capital A sine of, omega t plus phi, the amplitude is five centimetres, and omega is four pi radians per second; rad is radians, and the little minus one means per. Okay? So the period is two pi over four pi, which gives zero point five seconds. And the frequency is one over zero point five, which is two hertz. Right? Now be careful with the initial phase, because many students write the whole bracket. That bracket is the phase, which changes with time. The initial phase is its value at t equals zero, just pi over three. Clear? Finally, x of zero is five times sine sixty degrees, the small circle meaning degrees. That is five root three over two, approximately four point three three centimetres; the wavy equals sign means approximately. Got it?

6:28Example 2: SHM equation from x = 1 cm moving towards +x

Now a slightly harder one. Write the equation of an SHM of amplitude two centimetres and period four seconds, if at t equals zero the particle is at x equals one centimetre, moving towards plus x. Pause the video and try it first; watch the direction. Done? Okay, let's check. Omega equals two pi over capital T, that is two pi over four, which gives pi over two radians per second. Okay? So x equals two times sine of, pi t over two, plus phi. At t equals zero, x is one, so two sine phi equals one. Right? So sine phi is one half, and two angles give this: pi over six, and five pi over six. Which one is correct here? Both angles fit the position, so the direction decides. Moving towards plus x means x is increasing at t equals zero, and the sine of an angle is increasing when the cosine of that angle is positive. So cos phi must be greater than zero. Only pi over six does that; the double arrow means implies. With me? So x equals two times sine of, pi t over two, plus pi over six, in centimetres. At t equals zero, two sine pi over six is one, so it matches. Always check the direction, not just the position. That's it!

7:53Quick revision: restoring force, period, SHM or periodic

So, quick revision. SHM: acceleration towards the mean position, a equals minus omega squared times x, and F equals minus k times x. Then x equals capital A sine of, omega t plus phi, and the period, two pi root m over k, has no amplitude in it. And one frequency only: sine squared omega t is SHM, sine omega t plus sine two omega t is not. Now try the practice problems on this page. All the best!