A uniform rod of length 2.0 m is suspended through its endpoint about which it performs small angular oscillations in the vertical plane, its time period is nearly

Options

A. 1.6s
B. 1.8 s
C. 2.0 s
D. 2.3 s

Answer

To find the time period of a uniform rod of length 2.0 m suspended at one end and performing small angular oscillations, we can use the formula for the time period \( T \) of a physical pendulum:

\[ T = 2\pi \sqrt{\frac{I}{mgh}} \]

Where:
- \( I \) is the moment of inertia of the rod about the pivot point,
- \( m \) is the mass of the rod,
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the distance from the pivot to the center of mass of the rod.

### Step 1: Calculate the Moment of Inertia \( I \)

For a uniform rod of length \( L \) pivoted at one end, the moment of inertia \( I \) is given by:

\[ I = \frac{1}{3} m L^2 \]

Given that the length \( L = 2.0 \, \text{m} \):

\[ I = \frac{1}{3} m (2.0)^2 = \frac{1}{3} m \cdot 4 = \frac{4}{3} m \]

### Step 2: Determine the Distance to the Center of Mass \( h \)

The center of mass of a uniform rod is located at a distance of \( \frac{L}{2} \) from the pivot point. Therefore, for a rod of length \( L = 2.0 \, \text{m} \):

\[ h = \frac{L}{2} = \frac{2.0}{2} = 1.0 \, \text{m} \]

### Step 3: Substitute Values into the Time Period Formula

Now, substituting \( I \) and \( h \) into the time period formula:

\[ T = 2\pi \sqrt{\frac{I}{mgh}} = 2\pi \sqrt{\frac{\frac{4}{3} m}{mg \cdot 1.0}} \]

The mass \( m \) cancels out:

\[ T = 2\pi \sqrt{\frac{\frac{4}{3}}{g}} \]

Substituting \( g \approx 9.81 \, \text{m/s}^2 \):

\[ T = 2\pi \sqrt{\frac{4}{3 \cdot 9.81}} = 2\pi \sqrt{\frac{4}{29.43}} = 2\pi \sqrt{0.135} \]

### Step 4: Calculate the Numerical Value

Calculating \( \sqrt{0.135} \):

\[ \sqrt{0.135} \approx 0.367 \]

Now substituting back:

\[ T \approx 2\pi \cdot 0.367 \approx 2 \cdot 3.14 \cdot 0.367 \approx 2.28 \, \text{seconds} \]

### Final Result

Thus, the time period \( T \) is approximately:

\[ T \approx 2.3 \, \text{seconds} \]

### Answer:

The time period is nearly **2.3 seconds**.

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