First slide

Simple hormonic motion

Question

Function $x = A \sin^{2} ωt + B \cos^{2} ωt + C \sin ωt \cos ωt$ represents SHM :
(i) for any value of A, B and C (except C = 0)
(ii) if A=- B; C = 2B, amplitude = $|B \sqrt{2}|$
(iii) if A = B; C = 0
(iv) if A = B; C = 2B, amplitude $= |B|$

Easy

A

(i, ii, iv)
B

(ii, iii, iv)
C

(i, iii, iv)
D

(ii, iii)

Solution

$x = A \sin^{2} ωt + B \cos^{2} ωt + C sinωt cosωt x = \frac{A}{2} (1 - \cos 2 ωt) + \frac{B}{2} (1 + \cos 2 ωt) + \frac{C}{2} \sin 2 ωt$
For (a) : For any value of A, B and C (except C = 0), it represents SHM
For (b) : If A = -B and C = 2B, , x = B cos $ω$ t + B sin 2 $ω$ t which is SHM.
For (c ): If A=B,C =0, x= A = constant $\Rightarrow$ Not a SHM.
For (d) : If A=B,C =2B, x = B+B sin $2 ωt$ which is SHM.

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