Relation between Time Period, Restoring Couple, and Moment of Inertia

The time period of a suspended rigid body is influenced by two main factors: the restoring couple per unit twist of the suspension wire and the moment of inertia of the rigid body. Let's explore the relation between these variables using dimensions.

Restoring Couple per Unit Twist of Suspension Wire

The restoring couple per unit twist of the suspension wire can be denoted by the symbol k, and its dimensions can be represented as [M L^2 T^-2 θ^-1], where M represents mass, L represents length, T represents time, and θ represents angle.

Moment of Inertia of Rigid Body

The moment of inertia of a rigid body can be denoted by the symbol I, and its dimensions can be represented as [M L^2], where M represents mass and L represents length.

Time Period

The time period of a suspended rigid body can be denoted by the symbol T, and its dimensions can be represented as [T].

Relation Using Dimensions

To find the relation between the time period, restoring couple per unit twist, and moment of inertia, we can set up an equation using the dimensions:

T = k^a * I^b

We need to determine the values of exponents a and b.

By equating the dimensions on both sides of the equation, we get:

[T] = [M L^2 T^-2 θ^-1]^a * [M L^2]^b

Simplifying the equation, we get:

[T] = [M^(a+b) L^(2a+2b) T^(-2a) θ^(-a)]

Equating the dimensions individually, we have:

a + b = 0 (for mass dimension)
2a + 2b = 2 (for length dimension)
-2a = -2 (for time dimension)
-a = 0 (for angle dimension)

From the fourth equation, we get a = 0. Substituting this value in the third equation, we get b = 0.

Therefore, the relation between the time period, restoring couple per unit twist, and moment of inertia is:

T = k^0 * I^0

Simplifying further, we find that the time period is independent of both the restoring couple per unit twist and the moment of inertia of the rigid body.

Conclusion

The time period of a suspended rigid body is not influenced by the restoring couple per unit twist of the suspension wire or the moment of inertia of the rigid body. This implies that the time period remains constant regardless of changes in these variables.