On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is/are not correct.
a. \(y = asin ~2\pi t / T\)
b. \(y = a~sin~vt\)
c. \(y = {a \over T} sin ({t \over a})\)
d. \(y = a \sqrt 2 (sin {2 \pi t \over T} - cos {2 \pi t \over T})\)
(Symbols have their usual meanings.)
Choose the correct option:
1. (a, c)
2. (a, b)
3. (b, c)
4. (a, d)
Step: Find the dimensions for each option.
Now, by using the principle of homogeneity of dimensions LHS and RHS. of (a) and (b) will be the same, and is L
For the option (c), [LHS] = L
\(\begin{aligned} & [\mathrm{RHS}]=\frac{\mathrm{L}}{\mathrm{T}}=\mathrm{LT}^{-1} \\ & {[\mathrm{LHS}] \neq[\mathrm{RHS}]} \end{aligned}\)
Hence, (c) is not the correct option.
In option (b) dimension of the angle is vt i.e., L
\(\begin{array}{cc} \Rightarrow & \text { RHS }=L \cdot L=L^2 \text { and } L H S=L \\ \Rightarrow & \text { LHS } \neq \text { RHS } \end{array}\)
So, option (b) is also not correct.
Therefore, options b and c are not correct.
Hence, option (3) is the correct answer.
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