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The percentage error in measurement of a physical quantity $m$ given by $m = \pi \tan \theta $ is minimum when $\theta $ $=$ .......... $^o$ (Assume that error in $\theta $ remain constant)
$45$
$90$
$60$
$30$
Solution
$ m = k \tan \theta$
$dm = k \sec ^{2} \theta d \theta$
$\Rightarrow \frac{ dm }{ m }=\frac{ k \sec ^{2} \theta}{ k \tan \theta} d \theta$
$\Rightarrow \frac{ dm }{ m }=\frac{ d \theta}{\sin \theta \cos \theta}=\frac{2 d \theta}{\sin 2 \theta}$
$\Rightarrow \%$ error is minimum when $\sin 2 \theta$ has maximum value
$2 \theta=\frac{\pi}{2}$ or $\theta=45^{\circ}$
Similar Questions
A student determined Young's Modulus of elasticity using the formula $Y=\frac{M g L^{3}}{4 b d^{3} \delta} .$ The value of $g$ is taken to be $9.8 \,{m} / {s}^{2}$, without any significant error, his observation are as following.
| Physical Quantity | Least count of the Equipment used for measurement | Observed value |
| Mass $({M})$ | $1\; {g}$ | $2\; {kg}$ |
| Length of bar $(L)$ | $1\; {mm}$ | $1 \;{m}$ |
| Breadth of bar $(b)$ | $0.1\; {mm}$ | $4\; {cm}$ |
| Thickness of bar $(d)$ | $0.01\; {mm}$ | $0.4 \;{cm}$ |
| Depression $(\delta)$ | $0.01\; {mm}$ | $5 \;{mm}$ |
Then the fractional error in the measurement of ${Y}$ is
