Maharashtra Board Class 12 Physics Chapter 5 Oscillations PDF Download

5. Oscillations

Can You Recall?

1. What do you mean by linear motion and angular motion?

2. Can you give some practical examples of oscillations in our daily life?

3. What do you know about restoring force?

4. All musical instruments make use of oscillations, can you identify, where?

5. Why does a ball floating on water bobs up and down, if pushed down and released?

5.1 Introduction

Oscillation is a very common and interesting phenomenon in the world of Physics. In our daily life we come across various examples of oscillatory motion, like rocking of a cradle, swinging of a swing, motion of the pendulum of a clock, the vibrations of a guitar or violin string, up and down motion of the needle of a sewing machine, the motion of the prongs of a vibrating tuning fork, oscillations of a spring, etc. In these cases, the motion repeated after a certain interval of time is a periodic motion. Here the motion of an object is mostly to and fro or up and down.

Oscillatory motion is a periodic motion. In this chapter, we shall see that the displacement, velocity and acceleration for this motion can be represented by sine and cosine functions. These functions are known as harmonic functions. Therefore, an oscillatory motion obeying such functions is called harmonic motion. After studying this chapter, you will be able to understand the use of appropriate terminology to describe oscillations, simple harmonic motion (S.H.M.), graphical representations of S.H.M., energy changes during S.H.M., damping of oscillations, resonance, etc.

Teacher's Note

When a child swings on a swing in a park, this is oscillatory motion. The swing moves back and forth in the same path every time.

Exam Trick

Remember: Oscillatory motion = back and forth motion in the same path. Like a swing going to and fro.

Points to Remember

Oscillatory motion is periodic motion.
In oscillatory motion, an object moves back and forth along the same path.
Periodic motion is any motion that repeats after a fixed time.
Not all periodic motion is oscillatory motion.

5.2 Explanation of Periodic Motion

Any motion which repeats itself after a definite interval of time is called periodic motion. A body performing periodic motion goes on repeating the same set of movements. The time taken for one such set of movements is called its period or periodic time. At the end of each set of movements, the state of the body is the same as that at the beginning. Some examples of periodic motion are the motion of the moon around the earth and the motion of other planets around the sun, the motion of electrons around the nucleus, etc. As seen in Chapter 1, the uniform circular motion of any object is thus a periodic motion.

Another type of periodic motion in which a particle repeatedly moves to and fro along the same path is the oscillatory or vibratory motion. Every oscillatory motion is periodic but every periodic motion need not be oscillatory. Circular motion is periodic but it is not oscillatory.

The simplest form of oscillatory periodic motion is the simple harmonic motion in which every particle of the oscillating body moves to and fro, about its mean position, along a certain fixed path. If the path is a straight line, the motion is called linear simple harmonic motion and if the path is an arc of a circle, it is called angular simple harmonic motion. The smallest interval of time after which the to and fro motion is repeated is called its period (T) and the number of oscillations completed per unit time is called the frequency (n) of the periodic motion.

Can You Tell?

Is the motion of a leaf of a tree blowing in the wind periodic?

Teacher's Note

The Earth revolves around the Sun in a periodic motion. It takes the same time every year to complete one orbit.

Exam Trick

Remember: Period = time for one motion. Frequency = number of motions in one second. They are opposites of each other.

Points to Remember

Periodic motion repeats after a fixed time interval.
Period is the time taken for one complete motion.
Frequency is the number of motions in one second.
Simple harmonic motion is the simplest type of oscillatory motion.

5.3 Linear Simple Harmonic Motion (S.H.M.)

Place a rectangular block on a smooth frictionless horizontal surface. Attach one end of a spring to a rigid wall and the other end to the block as shown in Fig. 5.1. Pull the block of mass m towards the right and release it. The block will begin its to and fro motion on either side of its equilibrium position. This motion is linear simple harmonic motion.

Fig. 5.1(b) shows the equilibrium position in which the spring exerts no force on the block. If the block is displaced towards the right from its equilibrium position, the force exerted by the spring on the block is directed towards the left [Fig. 5.1(a)]. On account of its elastic properties, the spring tends to regain its original shape and size and therefore it exerts a restoring force on the block. This is responsible to bring it back to the original position. This force is proportional to the displacement but its direction is opposite to that of the displacement. If x is the displacement, the restoring force f is given by,

\[f = -kx\] --- (5.1)

where, k is a constant that depends upon the elastic properties of the spring. It is called the force constant. The negative sign indicates that the force and displacement are oppositely directed.

If the block is displaced towards left from its equilibrium position, the force exerted by the spring on the block is directed towards the right and its magnitude is proportional to the displacement from the mean position. (Fig. 5.1(c))

Thus, f = - kx can be used as the equation of motion of the block.

Now if the block is released from the rightmost position, the restoring force exerted by the spring accelerates it towards its equilibrium position. The acceleration (a) of the block is given by,

\[a = \frac{f}{m} = -\frac{k}{m}x\] --- (5.2)

where, m is mass of the block. This shows that the acceleration is also proportional to the displacement and its direction is opposite to that of the displacement, i.e., the force and acceleration are both directed towards the mean or equilibrium position.

As the block moves towards the mean position, its speed starts increasing due to its acceleration, but its displacement from the mean position goes on decreasing. When the block returns to its mean position, the displacement and hence force and acceleration are zero. The speed of the block at the mean position becomes maximum and hence its kinetic energy attains its maximum value. Thus, the block does not stop at the mean position, but continues to move beyond the mean position towards the left. During this process, the spring is compressed and it exerts a restoring force on the block towards right. Once again, the force and displacement are oppositely directed. This opposing force retards the motion of the block, so that the speed goes on reducing and finally it becomes zero. This position is shown in Fig. 5.1(c). In this position the displacement from the mean position and restoring force are maximum. This force now accelerates the block towards the right, towards the equilibrium position. The process goes on repeating that causes the block to oscillate on either side of its equilibrium (mean) position. Such oscillatory motion along a straight path is called linear simple harmonic motion (S.H.M.). Linear S.H.M. is defined as the linear periodic motion of a body, in which force (or acceleration) is always directed towards the mean position and its magnitude is proportional to the displacement from the mean position.

Remember This

For such a motion, as a convention, we shall always measure the displacement from the mean position. Also, as the entire motion is along a single straight line, we need not use vector notation (only ± signs will be enough).

Use Your Brain Power

If there is friction between a block and the resting surface, how will it govern the motion of the block?

Remember This

A complete oscillation is when the object goes from one extreme to other and back to the initial position. The conditions required for simple harmonic motion are: Oscillation of the particle is about a fixed point. The net force or acceleration is always directed towards the fixed point. The particle comes back to the fixed point due to restoring force. Harmonic oscillation is that oscillation which can be expressed in terms of a single harmonic function, such as \(x = a \sin \omega t\) or \(x = a \cos \omega t\). Non-harmonic oscillation is that oscillation which cannot be expressed in terms of single harmonic function. It may be a combination of two or more harmonic oscillations such as \(x = a \sin \omega t + b \sin 2\omega t\), etc.

Activity

Some experiments described below can be performed in the classroom to demonstrate S.H.M. Try to write their equations.

(a) A hydrometer is immersed in a glass jar filled with water. In the equilibrium position it floats vertically in water. If it is slightly depressed and released, it bobs up and down performing linear S.H.M.

(b) A U-tube is filled with a sufficiently long column of mercury. Initially when both the arms of U tube are exposed to atmosphere, the level of mercury in both the arms is the same. Now, if the level of mercury in one of the arms is depressed slightly and released, the level of mercury in each arm starts moving up and down about the equilibrium position, performing linear S.H.M.

Teacher's Note

A simple pendulum swinging back and forth shows S.H.M. This is very easy to see in a grandfather clock where the pendulum swings the same way every time.

Exam Trick

Remember: S.H.M. means force is always opposite to displacement. Like pushing a ball towards you, it pushes back away from you.

Points to Remember

In S.H.M., force and displacement are always opposite in direction.
S.H.M. has a fixed equilibrium or mean position.
The object always tries to return to the mean position.
S.H.M. is the simplest type of oscillatory motion.

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