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Question

Based on dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:


A
y = a sin(2πtT)
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B
y = a sin vt
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C
y = aT sin(ta)
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D
y = a2 [sin(2πtT)cos(2πtT)]
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Solution

The correct option is C y = aT sin(ta)
On the basis of “Principle of homogeneity” equal dimension can be added, subtracted or put equal to each other.

As y is the displacement of the particle, its unit should be equal to length, a is the amplitude of SHM and trigonometric functions are dimensionless.

(a) y = a sin(2πtT)

Here (2πtT) is dimensionless and [y]=[a]

(b) y = a sin vt

Here [vt]=[L1]
So trigonometric function is not dimensionless, hence this relation is not correct for simple harmonic motion.

(c) y = aT sin(ta)
Here (ta) has dimensions so trigonometric function is not dimensionless, and [y][aT]

Hence this relation is not correct for simple harmonic motion.

(d) y = a2 [sin(2πtT)cos(2πtT)]

Here (2πtT) is dimensionless and [y]=[a]

Final Answer: (b),(c)


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