Question

If momentum M, acceleration A and volume V are the fundamental units, then the dimensional formula for the kinetic energy will be:
1. [P1 A1/2 V1/6]
2. [P2 A1/2 V1/6]
3. [P1 A1 V1/6]
4. [P1 A1/2 V1]

Answer 1

Correct Answer - Option 1 : [P1 A1/2 V1/6]

CONCEPT:

Dimensions

• Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.

​CALCULATION:

The dimension of kinetic energy is given as,

⇒ [KE] = [M¹ L² T^-2]     -----(1)

The dimension of momentum is given as,

⇒ [M] = [M1 L1 T-1]     -----(2)

The dimension of acceleration is given as,

⇒ [A] = [M0 L1 T-2]     -----(3)

The dimension of the volume is given as,

⇒ [P] = [M0 L3 T0]     -----(4)

The dimensional formula for the kinetic energy in terms of momentum P, acceleration A, and volume V is given as,

⇒ [KE] = [P^x A^y V^z]     -----(5)

By equation 1, equation 2, equation 3, equation 4, and equation 5,

⇒ [M1 L2 T-2] = [M1 L1 T-1]^x [M0 L1 T-2]^y [M0 L3 T0]^z

⇒ [M1 L2 T-2] = [Mx Lx+y+3z T-x-2y]     -----(6)

By equation 6,

⇒ x = 1

⇒ x + y + 3z = 2

⇒ -x - 2y = -2

So,

⇒ -x - 2y = -2

⇒ -1 - 2y = -2

⇒ y = 1/2

And,

⇒ z = 1/6

• Therefore, the dimensional formula for the kinetic energy in terms of momentum P, acceleration A, and volume V is given as,

⇒ [KE] = [P¹ A^1/2 V^1/6]

• Hence, option 1 is correct.