Question

A force F is given by F = at + bt², where t is time. What are the dimensions of a and b?
1. MLT^-1 and MLT⁰
2. MLT^-3 and MLT^-4 
3. MLT^-4 and MLT¹
4. None of these

Answer 1

Correct Answer - Option 2 : MLT^-3 and MLT^-4 

CONCEPT:

Principle of homogeneity of dimensions:

• According to this principle, a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both sides of the equation are the same.
• This principle is based on the fact that only the physical quantities of the same kind can be added, subtracted, or compared.
• Thus, velocity can be added to velocity but not to force.

​EXPLANATION:

• The dimension of the force is given as,

⇒ [F] = [M L T^-2]     -----(1)

Let the dimensional formula of a and b is given as,

⇒ [a] = [M^x L^y T^z]

⇒ [b] = [M^p L^q T^r]

The dimensional formula of at and bt² should be equal to the dimensional formula of F.

⇒ [a][t] = [F]

⇒ [M^x L^y T^z][T] = [MLT^-2]

⇒ [M^x L^y T^z+1] = [MLT^-2]     -----(2)

By eqauation 2,

⇒ x = 1

⇒ y = 1

⇒ z +1 = -2

⇒ z = -3

Similarly,

⇒ [b][t²] = [F]

⇒ [M^p L^q T^r][T²] = [MLT^-2]

⇒ [M^p L^q T^r+2] = [MLT^-2]     -----(3)

By equation 3,

⇒ p = 1

⇒ q = 1

⇒ r + 2 = -2

⇒ r = -4

⇒ [a] = [M^x L^y T^z] = [M L T^-3]

⇒ [b] = [M^p L^q T^r] = [M L T^-4]

• Hence, option 2 is correct.