If orbital velocity of a planet is given by `v = G^a M^b R^c`, then what is the value of `(2a+b-3c)/(3b)?` [Where G= gravitational constant, M = mass of planet, R = Radius of orbit]

Explanation

To solve the problem, we start with the given expression for the orbital velocity of a planet:

\[ v = G^a M^b R^c \]

We know from physics that the orbital velocity \( v \) can also be expressed as:

\[ v = \sqrt{\frac{GM}{R}} \]

This can be rewritten as:

\[ v = G^{1/2} M^{1/2} R^{-1/2} \]

Now, we can compare the powers of \( G \), \( M \), and \( R \) from both expressions.

1. From the expression \( v = G^a M^b R^c \):
- The power of \( G \) is \( a \)
- The power of \( M \) is \( b \)
- The power of \( R \) is \( c \)

2. From the expression \( v = G^{1/2} M^{1/2} R^{-1/2} \):
- The power of \( G \) is \( \frac{1}{2} \)
- The power of \( M \) is \( \frac{1}{2} \)
- The power of \( R \) is \( -\frac{1}{2} \)

Now, we can set up the equations based on the powers:

- \( a = \frac{1}{2} \)
- \( b = \frac{1}{2} \)
- \( c = -\frac{1}{2} \)

Next, we need to find the value of the expression:

\[ \frac{2a + b - 3c}{3b} \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ = \frac{2 \left(\frac{1}{2}\right) + \left(\frac{1}{2}\right) - 3 \left(-\frac{1}{2}\right)}{3 \left(\frac{1}{2}\right)} \]

Calculating the numerator:

\[ = \frac{1 + \frac{1}{2} + \frac{3}{2}}{3 \left(\frac{1}{2}\right)} \]

Simplifying the numerator:

\[ = \frac{1 + 1 + 3}{3 \left(\frac{1}{2}\right)} = \frac{5}{3 \left(\frac{1}{2}\right)} = \frac{5}{\frac{3}{2}} = \frac{5 \cdot 2}{3} = \frac{10}{3} \]

Now, simplifying the denominator:

\[ = 3 \cdot \frac{1}{2} = \frac{3}{2} \]

Putting it all together, we have:

\[ = \frac{5}{\frac{3}{2}} = \frac{5 \cdot 2}{3} = \frac{10}{3} \]

Finally, we can simplify:

\[ = 2 \]

Thus, the value of \( \frac{2a + b - 3c}{3b} \) is:

\[ \boxed{2} \]