9702 · 17.1Simple harmonic oscillations

Oscillation terms and phase difference

0:00Hang a mass on a spring, pull it down a little, and let go. It bounces up and down. Now tie a small weight to a piece of string and let it swing. It sways from side to side. These look like different things, but if you plotted the position of each one against time, you'd get the same smooth, repeating wave. That's because both follow the same rule about acceleration, and that one rule predicts where the object is, how fast it's moving, and how hard it's being pulled back, at any moment.

0:25In this video we'll learn the language used to describe oscillations, find the rule that defines simple harmonic motion, and use it to build the equations and graphs for displacement, velocity and acceleration. One quick reminder from circular motion. When an object moves round a circle, we describe how fast it turns using angular speed, omega. It's the angle swept out per second, measured in radians per second. One complete turn is two pi radians, and it takes one period, T.

0:52So omega equals two pi over T. And since frequency f is one over T, omega also equals two pi f. We'll use that same omega for oscillations, and anything involving omega means working in radians. An oscillation is a repeated back-and-forth motion about a fixed central position, called the equilibrium position. It's where the object would sit at rest if you left it alone. Displacement, x, is the distance of the object from the equilibrium position, in a particular direction.

1:21It's positive on one side of equilibrium and negative on the other. Amplitude, x-nought, is the maximum displacement from the equilibrium position. Notice that it's measured from the middle, not from one extreme to the other. If a mass swings between 3 centimetres above and 3 centimetres below equilibrium, the amplitude is 3 centimetres, not 6. The period, T, is the time taken for one complete oscillation. Complete means back where it started, moving in the same direction.

1:49Bottom to top is only half an oscillation. The frequency, f, is the number of complete oscillations per unit time. It's measured in hertz, where one hertz is one oscillation per second. Period and frequency are just reciprocals: f equals one over T. And the angular frequency, omega, is two pi times the frequency. Its unit is radians per second. Think of one full oscillation as worth two pi radians, like one turn of a circle.

2:14Putting that together, the period can be written in terms of either one. T equals one over f, and T equals two pi over omega. The last term is phase difference. Release two identical pendulums together, and they move in step. They're in phase. Release the second one a moment later, and it's always a little behind. Phase difference tells us how far through a cycle one oscillation is ahead of or behind another.

2:39We can measure it as a fraction of a cycle, or as an angle. Since one complete cycle is two pi radians, or 360 degrees: A quarter of a cycle is pi over 2 radians, or 90 degrees. Half a cycle is pi radians, or 180 degrees. Two oscillations half a cycle apart are said to be in antiphase: when one is at its maximum positive displacement, the other is at its maximum negative displacement.

3:03A whole cycle, two pi, puts them back in phase. If you know the time delay between the two, you can work it out directly. The phase difference in radians is two pi times the time delay divided by the period. For example, two oscillators both have a period of 0.80 seconds. One reaches its maximum 0.20 seconds after the other. The time delay is a quarter of the period, so the phase difference is a quarter of a cycle, which is pi over 2 radians.

Defining simple harmonic motion

3:31Now the central idea. Not every oscillation is simple harmonic. Simple harmonic motion, or S H M, is a special kind of oscillation defined by one condition: The acceleration is proportional to the displacement from a fixed point, and is always directed in the opposite direction to the displacement. In other words, the acceleration is always directed towards that fixed point, which is the equilibrium position. Physically: twice the displacement, twice the acceleration.

3:57And whichever side it's on, the acceleration points back towards the middle. Written as an equation, that condition is: a equals minus omega squared x. Omega squared is a positive constant for a given oscillator. A bigger omega squared means a stronger pull back, so a faster oscillation. And the minus sign is the whole point. It says the acceleration is in the opposite direction to the displacement. When x is positive, a is negative.

4:23When x is negative, a is positive. The object is always being accelerated back towards the middle. Without the minus sign, a positive displacement would give a positive acceleration, pushing the object further away, harder and harder. It would run away and never come back. The minus sign is what makes it a restoring effect. Picture one oscillation. At the top, at maximum displacement, the mass is at rest for an instant, but the acceleration is at its maximum, pointing down.

4:52It speeds up towards the middle. At equilibrium, the displacement is zero, so the acceleration is zero, and it's moving fastest. Past the middle, the acceleration points the other way, slowing it until it stops at the other extreme. So maximum displacement comes with zero speed and maximum acceleration. Zero displacement comes with maximum speed and zero acceleration. A mass on a spring is the classic example. Displace it by x from equilibrium, and by Hooke's law the extra spring force is minus k x, proportional to x and back towards equilibrium.

5:24Dividing by the mass, the acceleration is minus k over m, times x. Comparing that with a equals minus omega squared x, you can see that omega squared equals k over m. A stiffer spring, or a lighter mass, gives a faster oscillation. A simple pendulum is the other standard example, with a condition. The restoring effect comes from the component of the weight along the arc. For small angles, that's very nearly proportional to the displacement, so the motion is very nearly simple harmonic.

5:53For large swings, the proportionality breaks down. So a pendulum is only S H M for small amplitudes.

SHM equations and graphs

6:00a equals minus omega squared x tells us about acceleration. But usually we want to know where the object is at a given time. The displacement equation that obeys this rule is: x equals x-nought sine omega t. A sine wave repeats every two pi radians, and omega t goes up by two pi in one period, so the displacement repeats once per period. And sine never exceeds one, so the biggest displacement is x-nought, the amplitude.

6:28Here's where circular motion comes in. Imagine a peg on the rim of a turntable, turning at a steady angular speed omega. Shine a light from the side so the peg casts a shadow on a wall. The peg goes round in a circle, but its shadow just moves up and down in a straight line. At time t, the peg has turned through an angle omega t, so the height of the shadow is the radius times sine omega t.

6:49That shadow moves with simple harmonic motion. The omega in S H M is the angular speed of this imaginary circle, which is why it's called the angular frequency. One important detail. x equals x-nought sine omega t assumes that the displacement is zero at time zero. In other words, you start the clock as the object passes through equilibrium, moving in the positive direction. If instead you pull the mass to its maximum displacement and release it from rest at time zero, the displacement starts at x-nought, not zero.

7:20Then the correct equation is x equals x-nought cos omega t, since cos of zero is one. Same motion, different starting point for the clock. Always check where the object is at time zero. And now a warning that costs a lot of students marks. In sine omega t, the angle omega t is in radians. Your calculator must be in radian mode. If it's left in degrees, you'll get a completely wrong answer, with no warning at all.

7:45We'll see exactly how wrong in the worked example. Velocity is the rate of change of displacement, so it's the gradient of the displacement-time graph. Look at the sine curve. The gradient is steepest as the curve crosses zero, at equilibrium, and it's zero at the peaks and troughs, where the curve momentarily flattens out. That's a cosine shape. When x equals x-nought sine omega t, the velocity is: v equals v-nought cos omega t, where v-nought is the maximum speed, and it equals omega times x-nought.

8:16That makes sense from the turntable picture too. The peg moves round a circle of radius x-nought at angular speed omega, so its speed is omega times the radius. When the shadow is passing through the middle, the peg is moving straight up or down, parallel to the wall, so the shadow moves with the peg's full speed, omega x-nought. Often you're told where the object is, not the time.

8:37So we want velocity in terms of displacement. Start with the two equations. x over x-nought equals sine omega t. And v over v-nought, which is v over omega x-nought, equals cos omega t. Now use the fact that sine squared plus cos squared always equals one. Square both, add them, and you get: x squared over x-nought squared, plus v squared over omega squared x-nought squared, equals one. Multiply everything by omega squared x-nought squared, and rearrange, and you get v squared equals omega squared, times x-nought squared minus x squared.

9:11Taking the square root gives v equals plus or minus omega times the square root of x-nought squared minus x squared. Why plus or minus? Because at any given displacement, the object passes through that point twice in each cycle: once going one way, and once coming back. Same speed, opposite directions. The equation gives you the speed, and you decide the sign from which way the object is moving.

9:34Now check the two extremes. At the equilibrium position, x is zero, so v equals plus or minus omega x-nought. That's the maximum speed, v-nought, exactly as before. At the extremes, x equals plus or minus x-nought, so x-nought squared minus x squared is zero, and the speed is zero. That's the instant the object stops and turns round. So maximum speed at equilibrium, zero speed at maximum displacement, just as in our physical picture.

10:02Let's put the three quantities on graphs, one above the other, all against the same time axis. Displacement is x-nought sine omega t, a sine curve with height x-nought. Velocity is omega x-nought cos omega t, a cosine curve with height omega x-nought. Acceleration is minus omega squared x, which is minus omega squared x-nought sine omega t. That's an upside-down sine curve, with height omega squared x-nought. Now compare them using those vertical lines.

10:32Velocity reaches its peak a quarter of a cycle before displacement does. So velocity leads displacement by a quarter of a cycle, which is pi over 2 radians. Acceleration is exactly upside down compared with displacement. When displacement is at its positive maximum, acceleration is at its negative maximum. So acceleration and displacement are in antiphase, with a phase difference of pi radians. That's the minus sign in a equals minus omega squared x, showing up on the graph.

11:01There's one more graph that matters: acceleration plotted against displacement. Since a equals minus omega squared x, this graph is a straight line through the origin, with a negative gradient. The gradient is minus omega squared. The line only runs from minus x-nought to plus x-nought, because the object never goes beyond its amplitude. This graph is the most direct test for S H M. If acceleration against displacement is a straight line through the origin with a negative gradient, the motion is simple harmonic.

11:29And you can find omega from the gradient.

Worked example: spring oscillation

11:31Let's use all of this. A mass on a spring oscillates with simple harmonic motion. Its amplitude is 5.0 centimetres, and its frequency is 2.0 hertz. At time zero, the mass passes through its equilibrium position, moving in the positive direction. Find: the angular frequency and the period; the maximum speed and the maximum acceleration; the speed and acceleration when the displacement is 3.0 centimetres; and the displacement 0.10 seconds after time zero.

12:01First, convert the amplitude to metres: 5.0 centimetres is 0.050 metres. Angular frequency: omega equals two pi f, which is two pi times 2.0, which is 4 pi, about 12.6 radians per second. Period: T equals one over f, which is one over 2.0, so 0.50 seconds. As a check, two pi over omega is two pi over 4 pi, which is also 0.50 seconds. Maximum speed: v-nought equals omega x-nought, which is 4 pi times 0.050.

12:36That's about 0.63 metres per second. Maximum acceleration happens at maximum displacement, so its size is omega squared times x-nought. Omega squared is about 158, and 158 times 0.050 is about 7.9 metres per second squared. Now the speed at a displacement of 3.0 centimetres, which is 0.030 metres. Use v equals plus or minus omega times the square root of x-nought squared minus x squared. x-nought squared minus x squared is 0.050 squared minus 0.030 squared.

13:12In units of ten to the minus four, that's 25 minus 9, which is 16, so 0.0016. The square root of 0.0016 is 0.040. So the speed is 12.6 times 0.040, which is about 0.50 metres per second. It could be moving either way at this point, so the velocity is plus or minus 0.50. The acceleration at that point is minus omega squared x: minus 158 times 0.030, which is about minus 4.7 metres per second squared.

13:47The minus sign tells you it points back towards equilibrium. Finally, the displacement at 0.10 seconds. The mass starts at equilibrium, so we use x equals x-nought sine omega t. Omega t is 4 pi times 0.10, which is 0.4 pi, about 1.26. That's 1.26 radians. With your calculator in radian mode, sine of 1.26 is about 0.951. So x equals 0.050 times 0.951, which is about 0.048 metres, or 4.8 centimetres.

14:23That's close to the amplitude, which makes sense: 0.10 seconds is a fifth of a period, a bit less than the quarter period it takes to reach the top. Now watch what happens if your calculator is in degree mode. It treats 1.26 as 1.26 degrees, and sine of that is only about 0.022. You'd get a displacement of about 0.0011 metres, around one millimetre. That's more than forty times too small, and nothing on the calculator would warn you.

14:51As a check, at the same moment v equals v-nought cos omega t, which is 0.63 times cos 1.26, about 0.19 metres per second. Putting the unrounded displacement, 0.0476 metres, into the square-root formula gives the same 0.19. If you use the rounded 0.048 instead, you get about 0.18, which is only a rounding difference.

Common mistakes and graph question

15:17Let's go through the mistakes that cost the most marks. Mistake one: calculator in degrees. Every sine and cos in this topic takes an angle in radians. Switch to radian mode before you start. Mistake two: mixing up f and omega. Frequency is in hertz, oscillations per second. Angular frequency is two pi times bigger, in radians per second. Using f where the formula needs omega makes every answer wrong by a factor of two pi, or two pi squared for acceleration.

15:48Mistake three: thinking acceleration is biggest where the object is fastest. At equilibrium, the speed is maximum but the acceleration is zero, because there's no displacement. At the extremes, the object is momentarily at rest, but the acceleration is maximum. Zero velocity does not mean zero acceleration. Mistake four: using sine when the object starts at maximum displacement. x equals x-nought sine omega t only works if x is zero at time zero.

16:16If the object is released from the extreme, use cos. Mistake five: doubling the amplitude. Amplitude is measured from the equilibrium position to one extreme, not from one extreme to the other. Mistake six: assuming every oscillation is S H M. It needs acceleration proportional to displacement and directed towards equilibrium, and a pendulum only fits for small swings. Graph questions are very common in this topic. Here's a typical one.

16:44"The graph shows how the acceleration of an object varies with its displacement. The line runs from a displacement of minus 0.020 metres to plus 0.020 metres. At a displacement of plus 0.020 metres, the acceleration is minus 8.0 metres per second squared. Explain why the graph shows that the motion is simple harmonic. Then determine the period and the maximum speed." For the explanation, you need both parts of the definition.

17:13The line passes through the origin, so the acceleration is proportional to the displacement. And the gradient is negative, so the acceleration is in the opposite direction to the displacement. Saying only "it's a straight line" won't get full marks. Next, the gradient is minus 8.0 divided by 0.020, which is minus 400. The gradient equals minus omega squared, so omega squared is 400, and omega is 20 radians per second.

17:40The period is two pi over omega, which is two pi over 20, about 0.31 seconds. For the maximum speed, you need the amplitude. The line stops at plus and minus 0.020 metres, so that's the amplitude. v-nought equals omega x-nought, which is 20 times 0.020, giving 0.40 metres per second.

SHM recap

18:01Let's bring it all together. Displacement is measured from equilibrium, and the amplitude is its maximum. The period is one over f, or two pi over omega, where omega equals two pi f. Phase difference is how far one oscillation is ahead of another, where one full cycle is two pi radians. Simple harmonic motion happens when the acceleration is proportional to the displacement from a fixed point and directed in the opposite direction.

18:25That's a equals minus omega squared x, and the minus sign is what makes it a restoring motion. If the object starts at equilibrium, its displacement is x equals x-nought sine omega t. Its velocity is v equals v-nought cos omega t, with v-nought equal to omega x-nought. And at any displacement, the velocity is plus or minus omega times the square root of x-nought squared minus x squared: maximum at the centre, zero at the extremes.

18:53On graphs, velocity leads displacement by pi over 2, acceleration is in antiphase with displacement, and acceleration against displacement is a straight line through the origin with gradient minus omega squared. And always, always, keep your calculator in radians.