Dimensional formula for the universal gravitational constant:

The universal gravitational constant, denoted by the symbol G, is a fundamental physical constant that appears in the law of universal gravitation, which describes the force of gravity between two objects.

The dimensional formula for any physical quantity represents its dimensions in terms of the fundamental quantities of mass (M), length (L), and time (T). Let's determine the dimensional formula for the universal gravitational constant using these fundamental quantities.

Step 1: Understanding the law of universal gravitation:

The law of universal gravitation states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The mathematical expression for the gravitational force (F) between two objects with masses (m1 and m2) separated by a distance (r) is given by:

F = G * (m1 * m2) / r^2

Here, G is the universal gravitational constant.

Step 2: Analyzing the equation:

Let's analyze the equation to determine the dimensions of G.

- The gravitational force (F) has dimensions of force, which is given by MLT^-2 (mass * length * time^-2).
- The product of masses (m1 * m2) has dimensions of mass squared, i.e., M^2.
- The distance (r) has dimensions of length, which is L.

Step 3: Finding the dimensions of G:

By substituting the dimensions of force, mass squared, and length into the equation, we can determine the dimensions of G:

MLT^-2 = G * M^2 / L^2

Simplifying the equation, we get:

G = (MLT^-2 * L^2) / M^2

G = M^-1 * L^3 * T^-2

Therefore, the dimensional formula for the universal gravitational constant is M^-1L^3T^-2.

Conclusion:

The correct dimensional formula for the universal gravitational constant is option a) M^-1L^3T^-2. This formula represents that the universal gravitational constant has dimensions of mass raised to the power of -1, length raised to the power of 3, and time raised to the power of -2.