Dimensional Formula for Length

There are various fundamental units that can be used to express different physical quantities. In this question, we are given force (F), acceleration (a), and time (T) as the fundamental units. We need to determine the dimensional formula for length using these units.

We know that the dimensional formula for any physical quantity is expressed as [M^a L^b T^c], where M, L, and T represent the dimensions of mass, length, and time, respectively. The exponents a, b, and c represent the powers to which the fundamental units are raised to express the physical quantity.

Therefore, the dimensional formula for length using force (F), acceleration (a), and time (T) can be calculated as follows:

Option (a) [F^0 * a * T^2]

- The dimension of force (F) is [M L T^-2], which means that F^0 = 1 (since any quantity raised to the power of zero is equal to 1).
- The dimension of acceleration (a) is [L T^-2].
- The dimension of time (T) is [T].
- So, the dimensional formula for length using F, a, and T would be [1 * L * T^2 * T^-2] = [L].

Option (b) [F * a^0 * T^2]

- The dimension of force (F) is [M L T^-2].
- The dimension of acceleration (a) is [L T^-2], which means that a^0 = 1.
- The dimension of time (T) is [T].
- So, the dimensional formula for length using F, a, and T would be [M L T^-2 * 1 * T^2] = [M L T^0] = [M L].

Option (c) [F * a^2 * T^0]

- The dimension of force (F) is [M L T^-2].
- The dimension of acceleration (a) is [L T^-2], which means that a^2 = [L^2 T^-4].
- The dimension of time (T) is [T^0] = 1.
- So, the dimensional formula for length using F, a, and T would be [M L T^-2 * L^2 T^-4 * 1] = [M L T^2 * L^-2] = [L].

Therefore, the correct answer is option (c) [F * a^2 * T^0], which represents the dimensional formula for length using force (F), acceleration (a), and time (T).