A wheel is rotating about its axis at a constant angular velocity. If suddenly an object sticks to it on the rim, then its M.I. will:

Options

A. increase
B. decrease
C. remain unchanged
D. nothing can be said

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Condition
The wheel is rotating about its axis with a constant angular velocity. This means it has an initial moment of inertia (M.I.) which we will denote as \( I_i \).

**Hint:** Remember that the moment of inertia depends on the mass distribution relative to the axis of rotation.

### Step 2: Define the Moment of Inertia
The moment of inertia for a wheel (considering it as a ring for simplicity) is given by the formula:
\[ I_i = m \cdot r^2 \]
where \( m \) is the mass of the wheel and \( r \) is its radius.

**Hint:** The moment of inertia is a measure of how difficult it is to change the rotational motion of an object.

### Step 3: Introduce the Sticking Object
When an object of mass \( m \) sticks to the rim of the wheel, we need to calculate the new moment of inertia \( I_f \).

**Hint:** When an additional mass is added to a rotating body, it affects the overall moment of inertia.

### Step 4: Calculate the Final Moment of Inertia
The new moment of inertia after the mass sticks to the rim can be calculated as:
\[ I_f = I_i + m \cdot r^2 \]
Substituting the expression for \( I_i \):
\[ I_f = (m \cdot r^2) + (m \cdot r^2) = 2m \cdot r^2 \]

**Hint:** The moment of inertia of the added mass is calculated using the same formula, considering its distance from the axis of rotation.

### Step 5: Compare Initial and Final Moment of Inertia
From our calculations:
\[ I_f = 2 \cdot I_i \]
This shows that the final moment of inertia \( I_f \) is greater than the initial moment of inertia \( I_i \).

**Hint:** Comparing the initial and final moments of inertia helps us determine how the system's rotational characteristics change.

### Conclusion
Since the final moment of inertia is greater than the initial moment of inertia, we conclude that the moment of inertia will **increase** when the object sticks to the rim of the wheel.

**Final Answer:** The moment of inertia will **increase**.