Question
Download Solution PDFConsider a sphere of uniformly distributed mass of 1 kg/m3 and radius 1 m. Its moment of intertia about one diameter is:-
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
- Moment of inertia (I) is analogous to mass acts in linear motion.
- The Moment of inertia is rotational inertia for which torque (turning force) is required for angular acceleration.
- Moment of inertia is also known as the second moment of mass, rotational inertia, angular mass, or mass moment of inertia.
- The moment of inertia is an extensive property by which its magnitude and values depend on the extent of the object which means how mass is concentrated at a different distance from the axis of rotation.
- The moment of inertia of a body about any axis is equal to the sum of the moment of the inertia of the body about a parallel axis passing through its center of mass and the product of Its mass and the square of the distance between the two parallel axes.
- As per definition the unit and dimension of the moment of inertia is a product of mass and square of the distance so, mass × square of distance hence its unit is kg m-2 and dimensional quantity is [M1L-2T0].
Calculation:
Here given a sphere is a uniformly distributed mass throughout its axis for as 1 kg m-3 and the radius is about r = 1 m, because it's continuously distributed mass we can say that its density is ρ = 1 kg m-3.
For finding the mass(m) we know that mass is the product of density (ρ) and Volume (V).
∴ m = ρ × V
∴ m = 1 ×
∴ m =
→ For uniformly distributed sphere now finding a moment of inertia we know that
∴ I =
∴ I = 1.67 kg m2
So, the moment of inertia is about 1.67 kg m2 for the solid sphere.
Additional InformationComparison of Translational and Rotational motion
Linear Motion or Translation motion | Rotational Motion about a fixed axis |
Displacement (x) | Angular Displacement (θ) |
velocity (v = |
Angular velocity (ω = |
Acceleration ( a = |
Angular acceleration ( α = |
Mass (m) | Moment of inertia (I) |
Force (F = ma) | Torque ( τ = I⋅ α ) |
Work (W = F ⋅ x) | Work ( W = τ ⋅ Θ ) |
Kinetic energy (K = |
Kinetic energy(K = |
Power ( P = F ⋅ v) |
Power ( P = τ ⋅ ω ) |
Linear Momentum (p = mv) | Angular Momentum (L = I ⋅ ω) |
Last updated on May 5, 2025
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