# A force F is given by F = at + bt², where t is time. What are the dimensions of a and b?

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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

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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 addedsubtracted, 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] = [Mx Ly Tz]

⇒ [b] = [Mp Lq Tr]

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

⇒ [a][t] = [F]

⇒ [Mx Ly Tz][T] = [MLT-2]

⇒ [Mx Ly Tz+1] = [MLT-2]     -----(2)

By eqauation 2,

⇒ x = 1

⇒ y = 1

⇒ z +1 = -2

⇒ z = -3

Similarly,

⇒ [b][t2] = [F]

⇒ [Mp Lq Tr][T2] = [MLT-2]

⇒ [Mp Lq Tr+2] = [MLT-2]     -----(3)

By equation 3,

⇒ p = 1

⇒ q = 1

⇒ r + 2 = -2

⇒ r = -4

⇒ [a] = [Mx Ly Tz] = [M L T-3]

⇒ [b] = [Mp Lq Tr] = [M L T-4]

• Hence, option 2 is correct.