Question
If $\text{x = a}\cos\omega\text{t + b}\sin\omega\text{t},$ show that it represents S.H.M.
✓
Answer
$\text{x = a}\cos\omega\text{t + b}\sin\omega\text{t}$$\frac{\text{dx}}{\text{dt}}=-\text{a}\omega\sin\omega\text{t}+\text{b}\omega\cos\omega\text{t}$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{a}\cos\omega\text{t}-\text{b}\omega^2\sin\omega\text{t}$
$=-\omega^2(\text{a}\cos\omega\text{t}+\text{b}\sin\omega\text{t})$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
$\therefore$ It represents a S. H. M.
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{a}\cos\omega\text{t}-\text{b}\omega^2\sin\omega\text{t}$
$=-\omega^2(\text{a}\cos\omega\text{t}+\text{b}\sin\omega\text{t})$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
$\therefore$ It represents a S. H. M.
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