Question

0) A particle performing S.H.M. with a period of 6 sec
is at the positive e.p. At time t = 1 sec, the particle
is at a distance of 3 cm from this position. Find the
amplitude of motion.
​

Answer 1

Answer:

Given:

SHM particles has time period of 6 seconds. At time = 1 sec, it is at a distance of 3cm from mean position.

To find:

Amplitude of the SHM

Concept:

Let us consider that the equation of the SHM be :

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\boxed{ \bold{ \sf{ \red{x = a \sin( \omega t)}}}}$

Also assuming that the initial phase is zero , i.e the particle starts from equilibrium position.

Calculation:

Considering amplitude to be "a"

Putting in all the data available :

$\therefore \: x = a \sin( \omega t)$

$\implies \: 3 = a \sin( \dfrac{2\pi}{T } \times 1 )$

$\implies \: 3 = a \sin( \dfrac{2\pi}{6 } \times 1 )$

$\implies 3 = a \sin( \dfrac{\pi}{3} )$

$\implies 3 = a (\dfrac{ \sqrt{3} }{2} )$

$\implies \: a = 2 \sqrt{3} \: cm$

Final answer :

$\boxed{ \large{ \blue{amplitude = 2 \sqrt{3} cm}}}$

Answer 2

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \boxed{ \bold{ \huge{ \purple{Answer}}}}} \\ \\ \star \sf \: \bold{ \orange{Given}} \\ \\ \mapsto \sf \: time \: period \: (T) = 6 \: s \\ \\ \mapsto \sf \: displacement \: from \: e.p. \: (x) = 3 \: cm \\ \\ \mapsto \sf \: time \: require \: (t) = 1 \: s \\ \\ \star \sf \: \bold{ \orange{To \: Find}} \\ \\ \mapsto \sf \: amplitude \: of \: shm \: (a) \\ \\ \star \sf \: \bold{ \orange{Formula}} \\ \\ \mapsto \sf \: formula \: of \: displacement \: from \: e.p. \\ \sf \: in \: SHM \: is \: given \: as... \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{\boxed{ \bold{ \sf{ \blue{x = a \sin( \omega{t}) }}}}} \\ \\ \mapsto \sf \: formula \: of \: angular \: freq. \: is \: given \: as \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \boxed{ \blue{ \sf{ \bold{ \omega = \frac{2{ \pi}}{T} }}}}} \\ \\ \star \sf \: \bold{ \orange{Calculation}} \\ \\ \mapsto \sf \: x = a \sin( \frac{2{ \pi}}{T} \times t) \\ \\ \mapsto \sf \: 3 = a \sin( \frac{2\pi}{6} \times 1 ) \\ \\ \mapsto \sf \: 3 = a \sin( \frac{\pi}{3} ) \\ \\ \mapsto \sf \: a = \frac{3}{ \sin( \frac{\pi}{3} ) } = 2 \sqrt{3} \: cm \: ( \because \: \sin( \frac{\pi}{3}) = \frac{ \sqrt{3} }{2} ) \\ \\ \therefore \underline{ \boxed{ \sf{ \bold{ \red{Amplitude = 2 \sqrt{3} \: cm}}}}}$