A solid sphere and a ring have equal masses and equal radius of gyration. If the sphere is rotating about its diameter and ring about an axis passing through and perpendicular to its plane, then the ratio of radius is \(\sqrt{\frac{x}{2} }\) then find the value of x.
A solid sphere and a ring have equal masses and equal radius of gyration. If the sphere is rotating about its diameter and ring about an axis passing through and perpendicular to its plane, then the ratio of radius is \(\sqrt{\frac{x}{2} }\) then find the value of x.
Correct Answer: 5
Approach Solution - 1
\((\frac{2}{5})mR_1^2 = mK_1^2 and R_2^2 =K_2\)
\(K_1 = \sqrt{(\frac{2}{5})R_1}\)
\(K_2=R_2\)
\(K_1 = K_2\)
\(\sqrt{(\frac{2}{5})} R_1=R_2\)
\(\frac{R_1}{R_2}= \sqrt{\frac{5}{2}}\)
Therefore, the value of x is 5.
Approach Solution -2
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Concepts Used:
Moment of Inertia
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.
Moment of inertia mainly depends on the following three factors:
- The density of the material
- Shape and size of the body
- Axis of rotation
Formula:
In general form, the moment of inertia can be expressed as,
I = m × r²
Where,
I = Moment of inertia.
m = sum of the product of the mass.
r = distance from the axis of the rotation.
M¹ L² T° is the dimensional formula of the moment of inertia.
The equation for moment of inertia is given by,
I = I = ∑mi ri²
Methods to calculate Moment of Inertia:
To calculate the moment of inertia, we use two important theorems-
- Perpendicular axis theorem
- Parallel axis theorem