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 given information about the particle moving along a circular path. Here’s the step-by-step solution:

### Step 1: Understand the given information
We know that the radial acceleration \( a_r \) is proportional to \( t^2 \). This can be expressed mathematically as:
\[ a_r = c t^2 \]
where \( c \) is a constant.

### Step 2: Relate radial acceleration to velocity
In circular motion, the radial (or centripetal) acceleration is also given by the formula:
\[ a_r = \frac{v^2}{r} \]
where \( v \) is the linear velocity and \( r \) is the radius of the circular path.

### Step 3: Equate the two expressions for radial acceleration
From the above two equations, we can equate:
\[ c t^2 = \frac{v^2}{r} \]
Rearranging gives us:
\[ v^2 = c r t^2 \]

### Step 4: Find the expression for velocity
Taking the square root of both sides, we find the expression for velocity:
\[ v = \sqrt{c r} \cdot t \]
This shows that the velocity \( v \) is directly proportional to time \( t \).

### Step 5: Determine the tangential acceleration
The tangential acceleration \( a_z \) is defined as the rate of change of velocity with respect to time:
\[ a_z = \frac{dv}{dt} \]
Substituting the expression for \( v \):
\[ a_z = \frac{d}{dt}(\sqrt{c r} \cdot t) = \sqrt{c r} \cdot \frac{d}{dt}(t) = \sqrt{c r} \]
Since \( \sqrt{c r} \) is a constant (as both \( c \) and \( r \) are constants), we conclude that:
\[ a_z = \sqrt{c r} \]
is independent of time.

### Step 6: Analyze the options
Now we need to check which of the given options is independent of time:
1. \( a_r \cdot a_z \): \( a_r \) is proportional to \( t^2 \) (depends on time), while \( a_z \) is constant. Thus, this product depends on time.
2. \( a_z \): As derived, \( a_z \) is constant and independent of time. This is a correct option.
3. \( \frac{a_r}{a_z} \): \( a_r \) depends on time, while \( a_z \) is constant. Thus, this ratio depends on time.
4. \( \frac{a_r^2}{a_z} \): Since \( a_r \) is proportional to \( t^2 \), \( a_r^2 \) is proportional to \( t^4 \). Therefore, this expression depends on time.

### Conclusion
The only quantity that is independent of time is:
\[ \boxed{a_z} \]