Inertia Formula | Problems (And Solutions)

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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