For a particle moving along a circular path the radial acceleration `a_r ` is proportional to `t^2` (square of time) . If `a_z` is tangential acceleration which of the following is independent of time

Options

A. `a_r . a_z`
B. `a_z`
C. `a_r//a_z`
D. `((a_r)^2)/(a_z)`

Explanation

To solve the problem, we need to analyze the relationship between radial acceleration \( a_r \) and tangential acceleration \( a_z \) for a particle moving along a circular path.

### Step-by-Step Solution:

1. **Understanding Radial Acceleration**:
- The radial acceleration \( a_r \) is given to be proportional to \( t^2 \). This can be expressed mathematically as:
\[ a_r = k t^2 \]
where \( k \) is a constant.

2. **Relating Radial Acceleration to Velocity**:
- The radial acceleration can also be expressed in terms of velocity \( v \) and radius \( r \) of the circular path:
\[ a_r = \frac{v^2}{r} \]
- By equating the two expressions for \( a_r \):
\[ \frac{v^2}{r} = k t^2 \]
- Rearranging gives:
\[ v^2 = r k t^2 \]

3. **Finding Velocity**:
- Taking the square root of both sides, we find the velocity \( v \):
\[ v = \sqrt{r k} \cdot t \]

4. **Finding Tangential Acceleration**:
- The tangential acceleration \( a_z \) is defined as the time derivative of velocity:
\[ a_z = \frac{dv}{dt} \]
- Substituting the expression for \( v \):
\[ a_z = \frac{d}{dt} \left( \sqrt{r k} \cdot t \right) \]
- Since \( \sqrt{r k} \) is a constant, we have:
\[ a_z = \sqrt{r k} \cdot \frac{d}{dt}(t) = \sqrt{r k} \cdot 1 = \sqrt{r k} \]
- Thus, \( a_z \) is independent of time.

5. **Analyzing the Options**:
- Now we need to determine which of the given options is independent of time:
- **Option A**: \( a_r \cdot a_z \) - This is dependent on \( a_r \), which is time-dependent.
- **Option B**: \( a_z \) - This is constant and independent of time.
- **Option C**: \( \frac{a_r}{a_z} \) - This involves \( a_r \), which is time-dependent.
- **Option D**: \( \frac{a_r^2}{a_z} \) - This also involves \( a_r \), which is time-dependent.

6. **Conclusion**:
- The only option that is independent of time is **Option B: \( a_z \)**.

### Final Answer:
**Option B: \( a_z \)** is independent of time.