# Inertia formula

Moment of inertia, also known as rotational inertia, is a measure of an object’s resistance to rotational motion. It depends on the object’s mass distribution. The moment of inertia of a point mass rotating around an axis at a distance r is given by the formula I = m × r2, where m represents the mass of the object.

For objects with more complex shapes or mass distributions, different formulas are used to determine their moment of inertia. For instance, a solid cylinder’s moment of inertia is given by I = ½ (m × r2). Another example is the moment of inertia for a solid sphere, which is given by I = ⅖ (m × r2), where r represents the radius of the sphere. Other shapes have their own unique formulas for calculating moment of inertia.

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## Practice problems

### Problem #1

An object with a mass of 4 kg is rotating around an axis at a distance of 5 meters. Calculate the moment of inertia of the object.

Solution

Given data:

• Mass of the object, m = 4 kg
• Distance from the axis of rotation, r = 5 m
• Moment of inertia of the object, I = ?

Applying the formula:

• I = m × r2
• I = 4 × (5)2
• I = 4 × 25
• I = 100 kg m2

Therefore, the moment of inertia of the object is 100 kg m2.

### Problem #2

A solid cylinder with a mass of 6 kg is rotating about its central axis. The cylinder has a radius of 2 meters. Determine the moment of inertia of the cylinder.

Solution

Given data:

• Mass of the cylinder, m = 6 kg
• Distance from the axis of rotation, r = 2 m
• Moment of inertia of the cylinder, I = ?

Applying the formula, for moment of inertia of the solid cylinder:

• I = ½ (m × r2)
• I = ½ (6 × 22)
• I = ½ (6 × 4)
• I = ½ (24)
• I = 12 kg m2

Therefore, the moment of inertia of the cylinder is 12 kg m2.

### Problem #3

A solid sphere with a mass of 2 kg is rotating about its central axis. The sphere has a radius of 5 meters. Determine the moment of inertia of the sphere.

Solution

Given data:

• Mass of the sphere, m = 2 kg
• Distance from the axis of rotation, r = 5 m
• Moment of inertia of the sphere, I = ?

Applying the formula, for moment of inertia of the sphere:

• I = ⅖ (m × r2)
• I = ⅖ (2 × 52)
• I = ⅖ (2 × 25)
• I = ⅖ (50)
• I = 2 × 10
• I = 20 kg m2

Therefore, the moment of inertia of the sphere is 20 kg m2.

### Problem #4

Two balls, A and B, with masses of 2 kg and 5 kg respectively, are connected by a rod of length 5 m. The system rotates about the axis CD. The distances of balls A and B from the axis of rotation are 2 m and 3 m respectively. 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 1 from the axis of rotation, rA = 2 m
• Distance of the ball 2 from the axis of rotation, rB = 3 m
• Moment of inertia of the system, I = ?

Applying the formula, for moment of inertia of the sphere:

• 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.