Simple Harmonic Motion: Motion can take any path. Linear motion is the easiest motion to study and we have seen how equations of motion and Newton's laws of motion can help us to determine velocities, displacements, accelerations, force, inertia, etc. There is another type of motion called oscillatory motion or vibratory motion. This motion is periodic in nature. That is after a regular time interval, the motion repeats itself. Unlike linear or circular motion, the force responsible for the motion of a body undergoing SHM, varies in magnitude and direction; the variation is periodic in nature and hence the motion is also periodic in nature. There are many examples of oscillatory motions in our daily life. needle of a sewing machine goes up–and–down at a certain regular frequency the pendulum of a grandfather's clock goes to and fro, repeatedly a child on a swing in a park goes back and forth repeatedly a weight on a spring when disturbed goes in up and down motion continuously. A block on a spring. The spring is hung from a ceiling. Displace the block a bit and the block goes up and down. Other than the example of the motion of a sewing needle, all the above examples of oscillatory motions are simple harmonic in nature. A classic example of SHM is the motion of a leaf in a pond. If waves are generated in the pond, the leaf will bob up and down in a repeated manner. A similar example of SHM is a cork bobbing up and down, when placed in a beaker containing water. Let us now try and understand how SHM occurs and what conditions are necessary for an SHM. Simple harmonic motion Take a steel ball, hanging at the end of a spring. The spring is held firmly at the other end. Let the spring along with the metal ball be kept steady at an equilibrium position (A). Now pull the ball down, displace it by a distance x and release it. You will notice the steel ball in an up and down motion. This type of motion is called a simple harmonic motion (more precisely, it is called linear simple harmonic motion). The word harmonic implies that the motion is periodic in nature. Periodicity shows that the ball returns to the same position after an interval of a definite time. If we analyze the motion carefully, we will see that when the metal ball is pulled down by distance x, the spring is stretched by a distance x. Because the spring is elastic in nature, it will try to restore its original shape or position. If we attach a scale to the vertical motion of the bob, we will see that at A, x = 0, at the lowest level B, x = xmax and at the highest level C, x = –xmax. The steel ball undergoes motion from xmax to –xmax. Let F be the restoring force. The restoring force has its origin in the potential energy of the spring material. It is easy to understand that the restoring force is proportional to the displacement x and acts in the opposite direction. This means that larger  the displacement, larger will be the restoring force, which will try and bring back the metal ball to its equilibrium position. F µX That is F = – k x The constant of proportionality k is called the force constant (or spring constant). Its value depends on the elastic properties of the spring, friction with the air molecules, etc. The force constant is equal to the restoring force F per unit displacement. The force or spring constant is measured in units of newtons per meter. The negative sign indicates that direction in which F and x act, is opposite to each other. To write in a vectorial form F = –k x When the metal ball is released, the restoring force will accelerate it towards its original equilibrium position. If m is the mass of the metal ball,  the acceleration can be calculated from the equation F = m.a Comparing the two equations, we can say that a = –kx/m Thus the acceleration of the object is proportional to its displacement but its direction is opposite to that of the direction of the displacement. The acceleration is directed towards the mean or the equilibrium position (which is x = 0 at point A). Once the metal ball is pulled and released, the restoring force will cause the ball to move back towards its original position. As this is happening, the acceleration will decrease continuously as x is decreasing continuously. When the metal ball reaches its original position, x = 0, both the force and the acceleration is zero. But the metal ball does not stop here!! Due to its momentum (or kinetic energy), the metal ball over–shoots its original equilibrium or mean position. Again a restoring force will act on the steel ball. The restoring force acts to stop the steel ball and bring it back to its mean position. Again at the mean position, the momentum of the ball will make to go down beyond its mean position. In this way the steel ball will bob up and down continuously, about its mean position. This type of motion is called a simple harmonic motion. A general definition of a simple harmonic motion is : The periodic motion of a body under the action of a force, which is always directed towards a fixed point and whose magnitude is proportional to the displacement of the body from the fixed point, is called simple harmonic motion. It has to be remembered that in a simple harmonic motion, the force that is acting on the body, varies both in magnitude and direction, and is periodic in nature. The force depends on the displacement from the mean position. The same can be said about the acceleration of the body. We can generalize and say that simple harmonic motion is that type of vibratory motion in which the restoring force acting on the vibrating body is directly proportional to its displacement. It should be noted that for a body to vibrate continuously and show periodic motion, three conditions are essential. There must be a restoring force to accelerate the body towards its mean position. The body must have inertia to keep itself moving beyond its mean position. The friction opposing the motion must be small. Keeping these three points in mind, let us discuss the simple pendulum and restoring forces in detail. Simple Pendulum A simple pendulum contains a bob made out of a small mass ball, attached to a thread that is supported rigidly at one end. The bob of the pendulum is free to swing. When the pendulum is at rest, it is in its mean position at A. The mean position is also the equilibrium position. At this position, all the forces are balanced and no net force is acting on the pendulum. Thus initially at A, the pendulum is stationary. Now if the bob of the pendulum is taken to one side, at position B and released, the bob will undergo back and forth motion. The motion will trace an arc of a circle whose radius is equal to the length of the pendulum. The bob will move towards position C through A. If there is no frictional force, the bob will move in the back and forth manner continuously, between the two extreme positions B and C. Here also, as in the case of spring and the bob, the restoring force Fand displacement x are opposed to each other. But the F exerted on bob it is due to gravitational acceleration g and mass of the bob alone. Some parameters associated with a simple pendulum: Length of the pendulum : The length of the thread from which the pendulum is hung, up to the centre of the bob, is called the length of the pendulum. This is denoted by L. It has to be borne in mind that the length of the thread plus the diameter of the bob does not constitute L. In fact L = length of the thread + radius of the bob. Oscillations of the pendulum : One complete to and fro motion is known as the oscillation of the pendulum. One oscillation is the motion taken for the bob to go from position B (initial position) through A (mean position) to C (the other extreme position) and back to B. One oscillation can also be defined as the motion taken for the bob to go from A to B to C (through A) and back to position A. It is denoted as w. Time period of the pendulum : The time taken for one oscillation by the pendulum is called the time period of the pendulum. Time period is also called period of the pendulum. This is denoted by T. Amplitude of the pendulum : The maximum displacement of the bob from its mean or equilibrium position is known as the amplitude of the pendulum. The straight–line length AB or AC is the amplitude of the pendulum. It has to be borne in mind that the amplitude is the straight line distance between points A and B or C and not the length of the arc AB or AC. Also the amplitude has to be reasonably smaller than the length of the pendulum. Amplitude is generally denoted as A. Frequency of the pendulum : The number of complete oscillations made a simple pendulum in one second is called its frequency. The SI unit for frequency is 'hertz' which is denoted by the symbol 'Hz'. Frequency is the reciprocal of time period (f=1/T). If we take a simple pendulum to a gravity free space (or zero gravity space), then it will not make any oscillations. This is because there will be no restoring force of gravity to keep the simple pendulum oscillating. So, the frequency of oscillation of a simple pendulum in a gravity free space will be zero. The air resistance encountered by the bob of the pendulum is assumed to be negligible in our discussions. Since a simple pendulum is a good example of simple harmonic motion, the study of the time period of the pendulum is important. Let us now see how the various parameters enumerated above affect the time period of the pendulum. Take a simple pendulum and measure the time period by varying the following parameters : Vary the length L of the pendulum. Vary the mass of the bob of the pendulum. Vary the material of the bob. Vary the amplitude of the pendulum. To measure the time period, take the bob of the pendulum to one side and release the bob gently. (It has to be noted that the amplitude of the pendulum has to be small). Let the pendulum swing for a short while. Start a stopwatch, when the bob reaches at one extreme position of its swing. Count a number of oscillations that the pendulum takes. Stop the stopwatch when the bob completes a certain number of oscillations. Let's say that you have measured time taken for the bob to complete 10 oscillations. The time period will be the time measured in seconds divided by 10. This number in seconds will give you the period of the pendulum in seconds. This procedure is necessary because it may not be possible to accurately calculate time taken for just one oscillation. This method gives an average and more accurate value of the time period. The time period of the pendulum: Depends on the length of the pendulum L. In fact T2 is proportional to L. Does not depend on the mass of the bob Does not depend on the material of the bob Does not depend on the amplitude of the pendulum (the amplitude should be small) Thus T2 µ L L = constant x T2 It has been calculated that the constant of proportionality is equal to acceleration due to gravity g divided by 4p2. The value of g is 9.8 m/s2. The left–hand side has units of meters per second square or m/s2. The units and the value of the constant is 0.248 m/s2. Since the time period of the pendulum is inversely proportional to square root of g, a sensitive pendulum is used to measure the variation of g on the surface of the earth. The value of g is very different on the surface of the earth and on other planets. On the earth g varies slightly in different locations, with an average value equal to 9.8 meters per second each second (9.8 m/s2). The gravitational acceleration g at a point on the earth, depends on the distance of that point from the centre of the earth. Since earth is oblate shaped, it is flattened at the poles and bulged at the equator, g is about 1% more at the poles than it is at the equator. Uses of simple pendulum The time period of a simple pendulum is very sensitive to length and the value of g. This is employed usefully for a number of applications. It is used as a clock. In olden days grandfather's clocks worked on the principle of the simple pendulum. It is used for measuring the acceleration due to gravity g on the earth and also on the moon and Mars. Since the time period dampens due to friction of the bob with the surrounding environment, a simple pendulum immersed in a liquid can measure the frictional force exerted by different liquids. This is then utilized for designing underwater vehicles, etc. It is used in many geological studies. The time period of a pendulum depends on the acceleration of gravity. Oil and mineral prospectors use very sensitive pendulums to detect slight differences in this acceleration. The acceleration due to gravity varies due to the variety of underlying formations. A sensitive pendulum is also used for underground movements of tectonic plates and is useful in the study of earthquakes. A simple pendulum may also be used to determine the latitude of a particular position on the earth. Difference between SHM of a bob placed at the end of a spring and a simple pendulum We have now seen two different simple harmonic motions, one of a bob at the end of a spring, and the other of a bob tied at the end of a string. The following table will give the difference between simple harmonic motion of a bob on a spring and a simple pendulum. (In all our discussions, we have considered the air resistance to motion of the bob to be negligible). Bob placed at the end of a string A simple pendulum It is an example of linear simple harmonic motion. Whether the spring is placed vertically or horizontally, the motion is the same. It is an example of angular simple harmonic motion. For all the discussions about its motion, the displacement of the bob from its mean position has to be small (small angular displacement). The restoring force is proportional to the spring constant k of the material of the spring. The restoring force is the component of g, acting in the direction of the mean position or gsinq The acceleration of the bob is not dependent on g. When placed vertically, the g does not affect the up and down motion, and when placed horizontally, the g experienced by the bob is same at all displacements. The acceleration of the bob is proportional to g The time period depends on the material of the spring. The time period is independent of the material of the string The time period depends on the mass of the bob. The time period is independent of the mass of the bob The time period is given by T = 2pÖ(k/m) The time period is given by T = 2p Ö(L/g)  Restoring Force We have seen how restoring force acts on a bob and is responsible for the simple harmonic motion. The restoring force is not a constant of motion and is proportional to the displacement of the bob. Let us have a little more careful look at the restoring force. The restoring force acting on an oscillating body (or vibrating body) is that force which tends to bring the body towards its mean position or central position. In other words, the force that acts towards the mean position or central position of an oscillating body is called a restoring force. The restoring force tries to restore the original position of an oscillating body. Let us take some examples to understand the meaning of restoring force more clearly. Example 1 The force exerted by a spring ( one end of which is fixed) on a ball attached at its free end is an example of restoring force. Consider a spring S with one end fixed and the other end carrying a ball A. When we displace the ball A to the right side by some distance to position B, the spring exerts an opposite restoring force which acts towards the centre position A of the ball. When we release the ball at B, this restoring force accelerates the ball until the ball reaches its centre position A. The inertia of the ball keeps it moving after it passes the centre point. Then the restoring force retards the ball until it reaches its extreme left position C. This process is repeated and the ball starts oscillating between B and C. It is very important to note here that the restoring force exerted by the spring is always directed towards the centre position (or mean position) of the vibrating ball. The force exerted by a stretched spring attached to a block of wood is also an example of restoring force. Example 2: The force exerted by the gravitational pull of the earth g, on a vibrating pendulum bob is an example of restoring force. Since the force acts towards the centre position of the bob and tries to restore its mean position, the restoring force arises due to gravitational pull of the earth. The direction of restoring force acting on the bob of an oscillating pendulum depends on the position of the bob. If the bob is at B, on the right side of its mean position, then the restoring force acts towards the left side. And if the bob is on the left side at C, then the restoring force acts towards the right side. Hence, the speed of bob always increases when it moves towards its centre position. The speed of bob decreases as it moves away from its centre position A,towards B or C. Why does a Pendulum oscillate A pendulum oscillates due to the combined effect of a restoring force, and the inertia possessed by the bob. This can be explained as follows. 1. When initially the bob is in the centre position A, two forces act on it, the gravitational force of earth W (or weight of the bob) acting downwards, and the tension T of thread acting upwards. In position A of the bob, the two forces acting on it are equal, opposite and along the same line, so they balance each other and the bob remains at rest. 2. When the bob is brought to position B to the right side and then released, then again the same two forces act on it. The gravitational force W acts along BW and the tension force T acts along BO. However, in position B, the two forces acting on the bob are not along the same line, so they cannot balance each other. Now, the gravitational force W acting on the bob in position B can be resolved into two components BD and BR, the component BD being in line with force of tension T and the component BR being at right angle to it. Only one component BD of the gravitational force is being balanced and neutralized by the upward tension T of thread OB. So, the other component BR of the gravitational force exerts a restoring force on the bob, which brings the bob back to its centre position A, to the left side. Since a restoring force acts on the bob in position B, the bob accelerates towards A and its speed goes on increasing. 3. When the accelerated bob reaches the central position A, due to its inertia, it does not stop at A, goes beyond the position A to the left side and reaches position C at the same distance on the left as B is on the right. But when the bob goes from position A towards C, then a restoring force acts on it and its speed gradually decreases till it becomes zero at position C. 4. When the bob is at position C, then again a restoring force CR causes it to accelerate towards the right side. Once again the bob crosses the centre position A due to its inertia and reaches position B. This process is repeated again and again due to which we can see the pendulum bob oscillating between points B and C. From the above discussion we conclude that the restoring force acting on the bob of an oscillating pendulum is that component of the gravitational force on the bob (or that component of the weight of the bob), which is not balanced by the force of tension in the pendulum thread. This restoring force is responsible for the pendulum oscillating to and fro. Of course, in all our discussions, we have assumed that the air resistance is negligible. In the presence of air resistance, the bob will lose its kinetic energy and eventually come to a halt. Acceleration and velocity of the bob in a SHM Whenever we have discussed motion, we have seen that force causes (change in) motion. Also equation of motion in terms of velocity, acceleration, displacement, mass is essential whenever we discuss motion. We can briefly calculate the equation of motion of the two SHM's we have discussed. For a bob at the end of a spring, acceleration a = –kx/m and velocity where K = spring constant, m = mass of the bob, w = frequency of oscillation, A = amplitude of oscillation and x = displacement from the equilibrium (mean) postion. Similarly, for a bob at the end of a string in a simple pendulum, acceleration is g along the horizontal displacement. a = – gsin θ, if θ is small, then sinθ = θ, θ is the angular displacement a = – gθ And velocity is given as A = amplitude of oscillation and θ = angular displacement from the equilibrium (mean) position. Types of Pendulums We have seen two types of pendulum. A spring pendulum which oscillates up and down and another a simple pendulum that goes back and forth. These oscillatory motions are due to restoring forces, which act opposite to the displacement of the pendulum bob. There are various types of pendulums. Some of them are discussed briefly below. Clock pendulums (or grandfather's clocks) They usually consist of a rod with a heavy weight at one end and a bearing at the other. A screw at the end of the rod permits the bob to be raised or lowered. When the bob is lowered, the pendulum swings more slowly, and the clock runs more slowly. When the bob is raised, the pendulum swings faster, and the clock runs faster. The bearing on which the pendulum swings must be as nearly frictionless as possible. It is often made of a knife–edge of agate set in a grooved agate plate. Clock Pendulum A device called an escapement is fastened to the mechanism of the clock. It gives small but regular pushes to the pendulum and keeps it swinging. The escapement lets one tooth of a toothed–wheel turn past it each time the pendulum swings aside. This action gives the clock its "ticktock" sound. The rod in a clock pendulum tends to expand when it is warmed and to shorten when it is cooled. If no correction is made, pendulum clocks will run slower in hot weather and faster in cold weather. Several means have been developed to make up for this effect. These pendulums consist of several brass and steel rods. The rods are arranged so that the brass rods raise the bob and the steel rods lower it as the temperature increases. Torsion pendulums Here a balanced set of weights are suspended on a wire. The wire becomes twisted like a spring as the weight rotates around it. Torsion pendulums are used in the detection of the effect of earth's rotation and revolution around the sun, whether such a motion is constant or accelerated or not. Bifilar pendulums Here two parallel wires hold the bob. They are pulled tight to provide a more controlled motion. A bifilar pendulum is sensitive to variations of the plumb line (line toward the earth's center of gravity). Bifilar pendulums have been used to show that the earth does not rotate on its axis at a constant speed. Such changes in the earth's rotation result from the pull of gravity exerted on the spinning earth by the sun and the moon. Foucault pendulum Jean Foucault hung a large iron ball on a wire about 200 feet (60 meters) long. With this pendulum, he showed that the earth rotates on its axis. A Foucault pendulum is similar to a simple pendulum, but its motion is not limited to a plane. The plane of its swing appears to change as the earth goes through its daily rotation. However, the pendulum actually continues to move in the plane in which it was set in motion, while the earth turns under it. At the equator, a Foucault pendulum does not change its apparent plane of swing. The change is one rotation every day at the North and South poles. Double pendulum One pendulum attached to the end of another pendulum is known as double pendulum. It is like holding a weight in your hand and swinging it with your shoulder and a flexible elbow. Unlike a simple pendulum, the equations for double pendulums do include the masses of the bobs. A new branch of Physics called Chaos is studied using double pendulums. Forced pendulum In this case, a force of specified amplitude, and frequency, is applied to a simple pendulum. The motion of the pendulum helps unravel repeated motions under externally applied force other than restoring force. Kater's pendulum It is used to measure g very accurately. It is made up of a sturdy rod with holes. The Kater's pendulum can be reversed for measuring its time period. It is hung on a knife–edge.