A long thin rod of length 2L rotates with a constant angular acceleration of `10rad//s^(2)` about an axis that is perpendicular to the rod and passes through its center.
What is the ratio of the tangential acceleration of a point on the end of the rod to that of a point a distance L/2 from the end of the rod?

Options

A. `1:1`
B. `2:1`
C. `1:4`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the tangential acceleration of two points on a rotating rod: one at the end of the rod and the other at a distance \( \frac{L}{2} \) from the end of the rod.

### Step-by-Step Solution:

1. **Identify the Length of the Rod**:
The length of the rod is given as \( 2L \).

2. **Determine the Angular Acceleration**:
The angular acceleration \( \alpha \) is given as \( 10 \, \text{rad/s}^2 \).

3. **Define the Tangential Acceleration Formula**:
The tangential acceleration \( a_t \) of a point at a distance \( r \) from the axis of rotation is given by the formula:
\[ a_t = r \cdot \alpha \]

4. **Calculate the Tangential Acceleration at the End of the Rod**:
For the point at the end of the rod (which is at a distance \( L \) from the center):
\[ a_{t1} = L \cdot \alpha \]
Substituting \( \alpha = 10 \, \text{rad/s}^2 \):
\[ a_{t1} = L \cdot 10 \]

5. **Calculate the Tangential Acceleration at the Point \( \frac{L}{2} \) from the End**:
The distance of this point from the center of the rod is \( L - \frac{L}{2} = \frac{L}{2} \):
\[ a_{t2} = \left(\frac{L}{2}\right) \cdot \alpha \]
Substituting \( \alpha = 10 \, \text{rad/s}^2 \):
\[ a_{t2} = \left(\frac{L}{2}\right) \cdot 10 \]

6. **Find the Ratio of the Tangential Accelerations**:
Now, we can find the ratio of the tangential accelerations:
\[ \text{Ratio} = \frac{a_{t1}}{a_{t2}} = \frac{L \cdot 10}{\left(\frac{L}{2}\right) \cdot 10} \]
Simplifying this:
\[ \text{Ratio} = \frac{L \cdot 10}{\frac{L}{2} \cdot 10} = \frac{L}{\frac{L}{2}} = 2 \]

7. **Final Result**:
Therefore, the ratio of the tangential acceleration of a point on the end of the rod to that of a point a distance \( \frac{L}{2} \) from the end of the rod is:
\[ \text{Ratio} = 2 : 1 \]