Question

The acceleration a of an object is given by the equation a=A+Bt+Ct^3 where t refers to time. (a) What are the dimensions of A, B, and C? (b) What are the SI units for the constants A, B, and C?

Answer 1

(a) A =
`[LT^(- 2)]`

B =
`[LT^(- 3)]`

`C = [LT^(- 5)]`

(b) A =
`ms^(- 2)`

B =
`ms^(- 3)`

C =
`ms^(- 5)`

Solution:

The acceleration of a body is the rate at which the velocity of the body changes.

Thus

`a = (\Delta v_(o))/(\Delta t)`

The SI unit of velocity of an object is
`ms^(- 1)` and its dimension is [LT^{- 1}] and for time, T the SI unit is second, s and dimension is [T] and hence

The SI unit and dimension for the acceleration of an object is
`ms^(- 2)` and [LT^{- 2}] respectively.

Now, as per the question:

acceleration, a =
`A + Bt + Ct^(3)`

(a) Now, according to the homogeneity principle in dimension, the dimensions on both the sides of the eqn must be equal,

For the above eqn:

`LT^(- 2) = A + Bt + Ct^(3)`

Thus the dimensions of :

A =
`[LT^(- 2)]`

BT =
`[LT^(- 2)]`

Thus for B

B =
`[LT^(- 3)]`

`CT^(3) = LT^(- 2)`

`C = [LT^(- 5)]`

(b) For the units of A, B and C, we will make use of their respective dimensional formula from part (a)

where

L corresponds to length in meter(m)

T corresponds to time in seconds(s)

Now, for:

A =
`[LT^(- 2)] = ms^(- 2)`

B =
`[LT^(- 3)] = ms^(- 3)`

C =
`[LT^(- 5)] = ms^(- 5)`