We are solving series of problems based on the concept of oscillations. It is also called vibratory or harmonic motion where there is to and fro motion that gets repeated at regular intervals of time. The vertical projection of uniform circular motion is simple harmonic and we have derived equations for displacement, velocity and acceleration for the body in simple harmonic motion based on that. The time taken to complete one oscillation is called time period and the number of oscillations per one second is called frequency.

**Problem**

A
body of known mass is connected with two springs and they in tern are connected
to rigid support as shown in the diagram below. We need to find the effective
time period of the system and the problem is as shown in the diagram below.

**Solution**

When
ever the body is slightly disturbed, it starts oscillating and one spring
expands and the other spring contracts by the same magnitude. As the force
acting on both of them is same, the two springs behaves as if like they are
connected in series and we can find the effective spring constant of the system
and time period as shown in the diagram below.

**Problem**

A
spring of spring constant K and length L is cut into two parts and the relation
between the lengths of two parts and their ratio is given to us in the problem
as shown in the diagram below. We need to measure the spring constant of one
part of the spring.

**Solution**

Spring
constant is the measure of nature of spring and it depends on the length of spring
in the inverse proportional ratio. Taking that into consideration, we need to
solve the problem as shown in the diagram below.

**Problem**

A
piece of wood known dimensions is given to us and its density is also known to
us as given in the problem below. It is floating in water with one surface
vertical to the surface. It is pushed down and released and it starts
oscillating. We need to measure the time period of the system.

**Solution**

When
ever a force is applied on the body, there is buoyant force also and the system
starts executing oscillatory motion with a certain restoring force. By
comparing that with the standard equation, we can solve the problem as shown in
the diagram below.

**Problem**

A
cylindrical piston is used to close a cylinder with a certain gas and when it
is slightly distributed, it starts oscillating in simple harmonic motion. We
need to find the time period of the system and the problem is as shown in the
diagram below.

**Solution**

By
comparing it with the standard equation, we can solve the problem as shown in
the diagram below. We need to identity the restoring force and the equation for
the acceleration of the body in SHM. The problem is solved as shown in the
diagram below. Here in the first case, we get the equation for the force.

We
need to equate to the product of mass and acceleration and we further need to
substitute the value of acceleration for a body in SHM so that we can compare
it with the standard equation. Thus we can get the time period of the system as
shown in the diagram below.

**Problem**

A
spherical ball of known mass and radius is rolling with out slipping on a
concave surface of known radius as shown in the diagram below. If the
oscillations are small, we need to find the time period of the system.

**Solution**

Let
us consider the spherical ball is at a particular position and we can find the
angle basing on the definition as shown in the diagram below. As we know the
equation for the acceleration of a body sliding on a inclined surface, by
writing that equation, we can get that and find the time period as shown in the
diagram below.

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