Physics, asked by anoop05maravi, 11 months ago

A body oscillates with shm according to the equation.
y(t)=25sin(2πt+π/4)
where 'x' is in metre and 't' is in second. Calculate
(i) displacement at time t=0
(ii) magnitude of maximum velocity

Answers

Answered by nirman95
45

Answer:

Given:

Equation of SHM is

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \boxed{ \green{ \bold{y = 25 \sin(2\pi t +  \dfrac{\pi}{4} )}}}

To find:

  • Displacement at t = 0
  • Magnitude of max Velocity

Concept:

We can very well understand that the body is already displaced at t = 0 seconds. This is because the SHM has an initial phase of π/4.

Max Velocity can be easily found out either by differentiation of the given Equation or by using formulas.

Calculation:

Putting t = 0 in the equation :

y = 25 \sin \{(2\pi  \times 0) +  \dfrac{\pi}{4}  \}

 =  > y = 25 \sin \{   \dfrac{\pi}{4}  \}

 =  > y = 25 \times  \dfrac{1}{ \sqrt{2} }

After rationalizing , we get :

 =  > y = 25 \sqrt{2}  \times  \dfrac{1}{ 2}

 =  > y = (12.5) \sqrt{2} \:  m

Max Velocity is calculated as:

 \boxed{ \blue{v \: max = a \times  \omega}}

 \implies v \: max = 25 \times  2\pi

 \implies v \: max = 50\pi \: m {s}^{ - 1}

Answered by Anonymous
35

 \underline{ \boxed{ \mathfrak{ \huge{ \orange{Answer}}}}} \\  \\ \star \rm \:  \blue{ Given} \\  \\  \rm equation \: of \: shm \:  \boxed{ \rm{\red{y(t) =  25 \sin(2\pi \: t +  \frac{\pi}{4} ) }}} \\  \\  \star \rm \:  \blue{To \: Find} \\  \\  \dagger \:  \rm \: displacement \:  at \: time \: t = 0 \\  \\  \dagger \rm \: magnitude \: of \: maximum \: velocity \\  \\  \star \rm \:  \blue{Formula} \\  \\  \rm formula \: of \: maximum \: velocity \: is \: given \: as \\  \\  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \underline{ \boxed{  \pink{ \mathfrak{v{ \tiny{max}} = a \omega}}}} \\  \\  \rm \: standard \: shm \: equation \\  \\   \:  \:  \:  \:  \:  \:  \:  \:  \: \underline{ \boxed{ \pink{ \mathfrak{{ \rm{y}}(t) = a \sin( \omega \: t +  \phi) }}}} \\  \\ \star \rm \:  \blue{Calculation} \\  \\  \dagger \rm \: for \: displacement \: at \: t = 0 \\  \\  \mapsto \rm  \: y(0) = 25 \sin( \frac{\pi}{4} )  = 25 \times  \frac{1}{ \sqrt{2} }  \\  \\  \mapsto \rm \boxed{  \rm{\orange{y(0) = 12.5\sqrt{2} \: m}} }\\  \\  \dagger \rm \: for \: maximum \: velocity \\  \\  \mapsto \rm \: comparing \: given \: equation \: with \: shm \: equation \\  \\  \mapsto \rm \:  \green{a = 25 \: m} \: and \:  \green{ \omega = 2\pi} \\  \\  \mapsto \rm \: v{ \tiny{max}} = 25 \times 2\pi \\  \\  \mapsto  \: \boxed{ \rm{ \orange{v{ \tiny{max}} = 50\pi \:  \frac{m}{s} }}}

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