Question

The displacement of two identical particle executing SHM are represented by equations x1=8 sin⁡ (10t+π/6) and x2=5 sin⁡ ωt.For what value of ω total energy of both the particle is same?

4 units

8 units

16 units

20 units

​

Answer 1

Displacement function of both the SHMs has been provided as follows :

$1) \: x1 = 8 \sin(10t + \frac{\pi}{6} )$

$2) \: x2 = 5 \sin( \omega t)$

We need to find the value of $\omega$ for which the total energy will be same for both SHMs.

We know that total energy for an SHM will be :

$\boxed{ \bold{TE = \frac{1}{2} m { \omega}^{2} {A}^{2} }}$

Comparison of TE for both SHMs , we can say that :

$\therefore \: \frac{1}{2} m { (\omega1)}^{2} {(A1)}^{2} = \frac{1}{2} m { (\omega2)}^{2} {(A2)}^{2}$

$= > {(10)}^{2} \times {(8)}^{2} = {( \omega2)}^{2} \times {(5)}^{2}$

$= > \omega2 = \dfrac{10 \times 8}{5}$

$= > \omega2 = 16$

So final answer :

Value of $\omega$ is 16 units

Answer 2

...

$\tt{\huge{\purple{ ..................... }}}$

QUESTION -

The displacement of two identical particle executing SHM are represented by equations x1=8 sin⁡ (10t+π/6) and x2=5 sin⁡ ωt.

For what value of ω total energy of both the particle is same?

[ A ] 4 units

[ B ] 8 units

[ C ] 16 units √√√√√√ [ Correct Answer ]

[ D ] 20 units

SOLUTION -

From the above Question, we are able to gather the following information...

The displacement of two identical particle executing SHM are represented by equations x1=8 sin⁡ (10t+π/6) and x2=5 sin⁡ ωt.

Since these particles are identical,

We can therefore state that the two energy levels of the two identical particles are equal.

So,

$\bold{ Energy \: Lvl _{1} = Energy \: Lvl _{2} }$

Displacement Equation Of Particle 1 :

=> x1 = 8 sin⁡ (10t+π/6)

Displacement Equation Of Particle 2 :

=> x2 = 5 sin⁡ ωt

_____________________________

FORMULAE USED :

We know that for a particle which is experiencing SHM,

Total Energy Level Of Particle :

$\tt{\purple{\leadsto{ \boxed{\boxed{ Total \: Energy \: Level = \dfrac{1}{2} . m . { \omega } ^ 2 . { A } ^ 2 }}}}}$

Now, The General Equation Of The Displacement Of A Particle Experiencing SHM -

$\tt{\orange{\mapsto{ \boxed{\boxed{ x(t) = A \cos ( 2 \pi f t ) }}}}}$

Now, from the above two equations, let us compare the equations to get the values of the Coefficients ω and A.

=> Comparing :

$\leadsto{ { \omega } _ { 1 } = 10 } \\ \\ \leadsto { A _ {1} = 8 }$

$\leadsto{ { \omega } _ { 2 } = ? } \\ \\ \leadsto { A _ {2} = 5 }$

Here we need to find ${ \omega _{2} }$

Now let us Substute these values into the energy lvl equation I defined earlier..

=> Substituting :

$\mapsto{ { 10} ^ 2 + { 8 } ^ 2 = { \omega _{2} } ^ 2 + { 5 } ^ 2 }$

Solving :

${ \omega _{2} } = 16$

___________________________________________

So the value of value of ω = 16, the energy level of both the particles are same.

Hence, Option C is The Correct Answer.

ANSWER :

Option C is The Correct Answer.

_____________________________________________