A simple pendulum consists of a single isolated bob suspended from a frictionless support by a light inextensible string. In equilibrium, the pendulum is held stationary in a vertical position (fig. 12.4) and its bob is at O. if the bob is disturbed from O to a point A, it will start moving towards its mean position under the action of one component of gravitational force. But at point O the velocity of the bob is maximum and due to inertia it passes by the mean position towards the other end. But now the velocity of the bob decreases, and becomes zero as it reaches at point B. the bob starts its motion from point B towards O and passes through O and reaches at point A. thus, the bob continues its motion in between the points A and B.
It is evident from the above description that as the speed increases while moving from point A to O, the acceleration of the bob is towards O. its acceleration remains towards point O during the motion from point O to B because now the speed decreases. It means, the acceleration to the bob is always directed towards the mean point O. we can prove mathematically that the acceleration of the bob is directly proportional to the displacement from equilibrium position. Hence, the characteristics of the motion. Therefore, we conclude that the motion of the simple pendulum is also simple harmonic motion.
Now we shall discuss some terms associated with simple harmonic motion. Point O is called mean position or equilibrium position whereas the point A or B is called extreme position the distance from point O to A or O to B is called amplitude of the vibration. During vibration between O to A and O to B is called amplitude of the bob from mean position O at any instant is called displacement of vibration. When the bob after moving from point A to point B comes back to A then it is said that the bob has completed one vibration. The time required to complete one vibration is called time period of the pendulum. It has been observed through experiments that the time period of the simple pendulum does not depend on its mass and amplitude. It can be proved mathematically that if the amplitude of the simple pendulum is not too large, then its time period T can be determined from the following equation:
Where l is the length of the pendulum which is equal to the distance between the point of suspension to the centre of the bob and g is the gravitational acceleration.
At position O, the bob is at the lowest position of the motion therefore, at this position its potential energy will be minimum and kinetic energy will be maximum. Hence, its velocity is maximum at the mean position. The bob is at highest level at position A or B, therefore, the potential energy will be maximum and kinetic energy will be minimum i.e., zero. In between the extreme and mean positions, at any point the energy of the bob is partly potential and partly kinetic; however, during vibration the total energy remains constant.
In simple harmonic motion, a body repeats it’s to and fro motion in equal intervals of time, about its fixed mean position. Now we can sum up the characteristics of S.H.M. as follows:
1. A body executing simple harmonic motion always vibrates about its position of equilibrium.
2. Its acceleration is always directed towards its mean position.
3. Its acceleration is directly proportional to its displacement from the mean position i.e., a = 0 at the mean position, and Amax, at the extreme positions.
4. Its velocity is maximum at the mean position and zero on the extreme positions.
In view of these characteristics we can define SHM as:
The acceleration of a body executing SHM is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position.
Category: 9th 10th