Spring–Mass Systems

Transcript

0:00Introduction

Hello everyone! Let's start with a small puzzle. Take a block on a spring, bouncing with some period, capital T, okay? Now we cut the spring into two equal halves, and attach both halves to the same block. Does the period stay the same, double, or become half? Many students say: it is the same spring, so the period stays the same. Hold on to your own guess. Spring problems come up often in JEE, so let's begin.

0:30The period of a block on a spring

So, pull a block of mass m by a distance x, and the spring pulls back with force capital F, equal to minus k times x, where k is the spring constant. The minus sign means the force points back towards the mean position, and it is proportional to x, the displacement. That is the condition for simple harmonic motion, or SHM, okay? So the period, capital T, is two pi times the square root of m over k. Pi is the Greek letter, and the tick-shaped sign is the square root. This holds for horizontal and vertical springs alike. Right? Now hang the spring from the ceiling, so gravity pulls the block down all the time. Let me ask you: does gravity change the period? The answer is no, it does not! Gravity only lowers the mean position, by x naught, which is x with a small subscript zero, equal to m times g, over k. There the spring force balances the weight. Clear? Now here is a topper's trick. Since m over k equals x naught over g, the period is also two pi times the square root of x naught over g. So from the stretch alone, you get the period straight away; it is the same period, written another way. Got it?

1:48Springs in parallel and in series

Now, two springs on one block, with spring constants k one and k two; the small one and two are subscripts. The figure shows three set-ups, and we will come to it in a moment. But first, springs side by side, called parallel. Both stretch by the same x, so their forces add, and k equals k one plus k two. Okay? Now look at part a of the figure: a block of mass m between two springs, each fixed to a wall. The springs sit in a line, one after the other. So this is series, right? That is exactly the trap, so be careful! Push the block right by x. The left spring stretches and pulls it back, and the right spring is squeezed and pushes it back. Both forces point the same way, so they add, and k is again k one plus k two. Clear? Now part b: springs joined end to end, called series. The same force passes through both springs, and their stretches add. So one over k equals one over k one, plus one over k two. With me? Take the common denominator, and k equals k one times k two, over k one plus k two. Part c, on the right, is a physical pendulum, for a later lesson. Right?

3:07Cutting a spring

Now let's cut a spring of length l and spring constant k into two pieces, with lengths in the ratio l one to l two. Let me ask you: is a shorter piece stiffer or softer than the whole spring? Stiffer, yes, exactly! For pieces of one spring, k times l stays constant. So the piece of length l one has spring constant k one, equal to k times l, over l one. The shorter the piece, the bigger its spring constant. Got it?

3:39Two blocks joined by a spring

Next, blocks of masses m one and m two, joined by a spring on a smooth floor, with no wall at all. Pull them apart, and both oscillate, okay? The period is two pi times the square root of mu over k. Mu, the Greek letter like a u with a tail, is the reduced mass: m one times m two, over m one plus m two. Do not use m one plus m two here. Right?

4:06Placing a mass on an oscillating block

Now here is one more twist. Take a block of mass capital M, on a spring on a smooth floor, oscillating with amplitude capital A, and gently place a mass, small m, on it, so the two move together, okay? Place it as the block passes the mean position, where it moves fastest. Momentum is conserved, and the new mass arrives with no sideways speed, so the pair slows down. The new amplitude, A prime, with a little dash, is capital A times the square root of capital M over capital M plus small m, which is smaller. Clear? Now place the mass at an extreme instead, where the block is at rest for a moment. What is the new amplitude now? It stays capital A, exactly! There is no speed, so no momentum to share, and the spring is still stretched, or squeezed, by capital A, so that point stays the extreme, okay? In both cases, the oscillating mass is now capital M plus small m. So the new period, T prime, is two pi times the square root of capital M plus small m, over k. With me?

5:13Example 1: 100 and 300 N/m springs in parallel and series

Alright, example one: springs of one hundred and three hundred newtons per metre carry a one kilogram mass, first in parallel, and then in series. Find the two periods. Capital N, m to the power minus one, means newtons per metre, and k g is kilograms. Pause the video and try it first. Done? Okay, let's check. In parallel the spring constants add: one hundred plus three hundred, which is four hundred newtons per metre. Right? So capital T is two pi times the square root of one over four hundred, which is two pi over twenty, or pi over ten. That is about zero point three one seconds; the wavy equals sign means approximately, and s is seconds. Clear? Now for series, tell me first: will the combined spring constant be more than one hundred, or less? It comes out less than one hundred, yes! One hundred times three hundred, over one hundred plus three hundred, is thirty thousand over four hundred, which gives seventy five newtons per metre. That is softer than either spring. Okay? So capital T is two pi times the square root of one over seventy five, about zero point seven three seconds. The softer combination gives the longer period. Got it?

6:34Example 2: spring cut in half, halves in parallel

Remember our puzzle from the start? Let's solve it. Example two: a block oscillates on a spring with a period called capital T, and the spring is then cut into two halves, both attached to the block in parallel. Find the new period. Pause the video and try it first. Done? Okay, let's check. Each half has length l over two. Since k times l stays constant, each half has spring constant k times l, over half of l, which is two k. Right? Both halves pull on the block in parallel, so their spring constants add: two k plus two k, which is four k. With me? So the new period, T prime, is two pi times the square root of m over four k. The four comes out of the root as two, so this is half times two pi root m over k, which is capital T over two. The period becomes half! Cutting changes each spring constant, and our puzzle is solved. Clear?

7:35Quick revision: T = 2 pi root m/k, series, parallel, cut

So, quick revision. Capital T is two pi root m over k, and gravity only shifts the mean position, by m times g, over k. In parallel, and for a block between two springs, the spring constants add; in series, their reciprocals add. When a spring is cut, k times l stays constant, and two blocks joined by a spring use the reduced mass mu. Now go and try the practice problems on this page. All the best!