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Question
On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct ______.
- y = `a sin (2πt)/T`
- y = `a sin vt`
- y = `a/T sin (t/a)`
- y = `asqrt(2) (sin (2pit)/T - cos (2pit)/T)`
Solution
b. y = `a sin vt`
c. y = `a/T sin (t/a)`
Explanation:
The argument of trigonometric functions (sin, cos etc.) should be dimensionless. y is displacement and according to the principle of homogeneity of dimensions LHS and RHS.
`[Y] = [L], [a] = [L]`
`[(2pit)/T] = ([T])/([T]) = [T^0]`
`[vt] = [v][t] = [LT^-1][T] = [L]`
`[a/T] = ([a])/([T]) = ([L])/([T]) = [LT^-1]`
`[t/a] = [L^-1T]`
[LHS] ≠ [RHS]
Hence, (c) is not the correct option.
=> LHS ≠ RHS.
So, option (b) is also not correct.
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