9. A student writes [tex]\sqrt{\frac{R}{2 G M}}[/tex] for escape velocity. Check the correctness of the formula by using dimensional analysis.



Answer :

Sure, let's analyze the correctness of the formula [tex]\(\sqrt{\frac{R}{2 G M}}\)[/tex] for escape velocity using dimensional analysis.

### Step-by-Step Dimensional Analysis:

1. Understand the Dimensions of Each Quantity:
- [tex]\( R \)[/tex] (radius) has the dimension of length [tex]\([L]\)[/tex].
- [tex]\( G \)[/tex] (gravitational constant) has the dimensions [tex]\([L^3 M^{-1} T^{-2}]\)[/tex].
- [tex]\( M \)[/tex] (mass) has the dimension [tex]\([M]\)[/tex].

2. Combine the Dimensions in the Given Formula:
The formula provided is [tex]\(\frac{R}{2 G M}\)[/tex]. We need to analyze the dimensions of this expression inside the square root:

[tex]\[ \frac{R}{2 G M} \][/tex]

3. Analyze the Individual Terms in the Denominator:
- The constant 2 is dimensionless.
- The dimension of [tex]\( G \)[/tex] is [tex]\([L^3 M^{-1} T^{-2}]\)[/tex].
- The dimension of [tex]\( M \)[/tex] is [tex]\([M]\)[/tex].

4. Combine the Dimensions in the Denominator:
Combine the dimensions of [tex]\( G \)[/tex] and [tex]\( M \)[/tex]:

[tex]\[ 2 \times G \times M = 2 \times [L^3 M^{-1} T^{-2}] \times [M] = [L^3 M^{-1} T^{-2}] \times [M] = [L^3 T^{-2}] \][/tex]

5. Evaluate the Dimensions of the Entire Expression:
Next, we evaluate the dimensions of the expression [tex]\( \frac{R}{2 G M} \)[/tex]:

[tex]\[ \frac{R}{2 G M} = \frac{[L]}{[L^3 T^{-2}]} = [L] \times [L^{-3} T^{2}] = [L^{-2} T^{2}] \][/tex]

6. Take the Square Root of the Dimensions:
Finally, take the square root of the dimensions of the expression:

[tex]\[ \sqrt{[L^{-2} T^2]} = [L^{-1} T] \][/tex]

### Conclusion:
The dimension of the expression [tex]\(\sqrt{\frac{R}{2 G M}}\)[/tex] is found to be [tex]\([L^{-1} T]\)[/tex].

For escape velocity, the correct dimension should be that of velocity, which is [tex]\([L T^{-1}]\)[/tex]. The dimension [tex]\([L^{-1} T]\)[/tex] does not match this, indicating that the formula [tex]\(\sqrt{\frac{R}{2 G M}}\)[/tex] is dimensionally incorrect for escape velocity.

Thus, the student’s formula [tex]\(\sqrt{\frac{R}{2 G M}}\)[/tex] is not correct for calculating escape velocity.