The M.I. of a disc about its diameter is 2 units. Its M.I. about axis through a point on its rim and in the plane of the disc isa)4 unitb)6 unitc)8 unitd)10 unitCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

We know that for a disc of mass m and radius r
MI of a disc about its diameter = mr²/4 = 2
And also MI about a point on its rim = mr²/4 + mr²
= 5mr²/4
= 5 x 2 = 10

Answer

Mi of the disc about it axis =(mr^2)/4 now mr^/4=2 so r^2=8/m now mi about a point on rim =mr^2/4+mr^2 =5mr^2/4 =(5m/4)(8/m) =10 units

Answer

We know that for a disc of mass m and radius r
MI of a disc about its diameter = mr²/4 = 2
And also MI about a point on its rim = mr²/4 + mr²
= 5mr²/4
= 5 x 2 = 10

Answer

The moment of inertia (M.I.) of an object is a measure of its resistance to rotational motion, specifically about a given axis. It depends on the mass distribution and the axis of rotation. In this case, we are given a disc with a known M.I. about its diameter and asked to find the M.I. about an axis through a point on its rim and in the plane of the disc.

1. Understanding the concept of Moment of Inertia:
- Moment of inertia is defined as the sum of the products of the mass of each particle in a body and the square of its perpendicular distance from the axis of rotation.
- It quantifies how the mass of an object is distributed around an axis of rotation.

2. Moment of Inertia about the Diameter:
- The moment of inertia of a disc about an axis passing through its diameter (also known as the central axis) is given by the formula I = (1/4) * m * r^2, where m is the mass of the disc and r is its radius.
- In this case, the given M.I. is 2 units.

3. Moment of Inertia about an Axis through a Point on the Rim:
- The moment of inertia about an axis passing through a point on the rim of a disc and in the plane of the disc can be calculated using the parallel axis theorem.
- The parallel axis theorem states that the M.I. about an axis parallel to and a distance 'd' away from an axis through the center of mass is given by I_parallel = I_cm + m * d^2, where I_parallel is the M.I. about the parallel axis, I_cm is the M.I. about the center of mass, m is the mass of the object, and d is the perpendicular distance between the two axes.
- In this case, the M.I. about the diameter is given as 2 units.

4. Applying the Parallel Axis Theorem:
- Since the M.I. about the diameter is given as 2 units, we can calculate the M.I. about the parallel axis using the parallel axis theorem.
- Considering that the M.I. about the center of mass of the disc is zero (since the axis is passing through the center), we can simplify the equation to I_parallel = m * d^2.
- The perpendicular distance 'd' in this case is equal to the radius of the disc, as the axis is passing through a point on the rim.
- Substituting the values, I_parallel = m * r^2.

5. Determining the Correct Option:
- Comparing the M.I. about the parallel axis (I_parallel) with the M.I. about the diameter (2 units), we can see that I_parallel = 2 * r^2.
- Thus, the M.I. about the parallel axis is 2 times the M.I. about the diameter.
- Since the M.I. about the diameter is given as 2 units, the M.I. about the parallel axis is 2 * 2 = 4 units.
- Therefore, the correct option is D) 10 units.

In summary, the moment of inertia about an axis passing through a point on the rim and in the plane of the disc is 4 units.

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