Inertia Formula | Problems (And Solutions)

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Inertia Formula

Moment of inertia or rotational inertia (I) is equal to the product of mass (m) and the square of distance (r). Using the formula of rotational inertia: I = m × r2, the value of rotational inertia of an object can be calculated.

Let’s solve some problems based on this formula, so you’ll get a clear idea.

Inertia Practice Problems

Problem 1: A disc having a mass of 4 kg is rotating about its center. If the distance from the axis of rotation is 5 m, then calculate the moment of inertia of a disc.

Solution:

Given data:
Mass of a disc, m = 4 kg
Distance from the axis of rotation, r = 5 m
Moment of inertia of a disc, I = ?

Using the formula of moment of inertia,
I = m × r2
I = 4 × (5)2
I = 4 × 25
I = 100 kg m2

Therefore, the moment of inertia of a disc is 100 kg m2.

Problem 2: Calculate the moment of inertia of a 250 gm ring rotating about its center. The distance from the axis of rotation is 6 m.

Solution:

Given data:
Moment of inertia of a ring, I = ?
Mass of a ring, m = 250 gm = 0.25 kg
Distance from the axis of rotation, r = 6 m

Using the formula of moment of inertia,
I = m × r2
I = 0.25 × (6)2
I = 0.25 × 36
I = 9 kg m2

Therefore, the moment of inertia of a ring is 9 kg m2.

Problem 3: Two balls A and B of mass 2 kg and 5 kg are connected by a rod of length 5 m and rotates about the axis CD. The distances of the two balls A and B from the axis of rotation are 2 m and 3 m. Calculate the moment of inertia of the system about the axis CD.

Solution:

Given data:
Mass of the ball A, mA = 2 kg
Mass of the ball B, mB = 5 kg
Distance of the ball A from the axis of rotation, rA = 2 m
Distance of the ball B from the axis of rotation, rB = 3 m
Moment of inertia of the system, I = ?

Using the formula of moment of inertia,
I = (mA × rA2) + (mB × rB2)
I = (2 × 22) + (5 × 32)
I = (2 × 4) + (5 × 9)
I = 8 + 45
I = 53 kg m2

Therefore, the moment of inertia of the system is 53 kg m2.

Problem 4: A ball of mass 300 gm is rotating on its own axis. One another ball of mass 400 gm is connected to it by the rod at a distance of 2 m. Calculate the moment of inertia of the system.

Solution:

Given data:
Mass of the ball 1, m1 = 300 gm = 0.3 kg
Mass of the ball 2, m2 = 400 gm = 0.4 kg
Distance of the ball 1 from the axis of rotation, r1 = 0 m
Distance of the ball 2 from the axis of rotation, r2 = 2 m
Moment of inertia of the system, I = ?

Using the formula of moment of inertia,
I = (m1 × r12) + (m2 × r22)
I = (0.3 × 02) + (0.4 × 22)
I = (0.3) + (0.4 × 4)
I = 0.3 + 1.6
I = 1.9 kg m2

Therefore, the moment of inertia of the system is 1.9 kg m2.

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