Question

The potential energy U of a particle is given
by U = 2.5x^2 + 100 joule. Is the motion
simple harmonic. If the mass of the particle
is 0.2 kg, what is its time period:

(1) Yes, 2.5 sec.
(2) Yes, 1.26 sec.
(3) Yes, 5.2 sec.
(4) No​

Answer 1

Answer

$\orange{\boxed{\boxed{\boxed{\pink{\underline{\underline{\red{\mathfrak{Given</p><p>}}}}}}}}}$

Potential energy = 2.5 x² + 100

$\orange{\boxed{\boxed{\boxed{\pink{\underline{\underline{\red{\mathfrak{To\:Find}}}}}}}}}$

If this Motion is SHM or not ??

If yes, then it's Time Period .

$\orange{\boxed{\boxed{\boxed{\pink{\underline{\underline{\red{\mathfrak{Concept}}}}}}}}}$

Using the Potential Energy equation, we can find out the Force equation by Differentiation.

Then we can compare it with the standard equation of SHM.

$\orange{\boxed{\boxed{\boxed{\pink{\underline{\underline{\red{\mathfrak{Differentiation}}}}}}}}}$

U = 2.5 x² + 100

=> F = - dU/dx = - 5x

=> F = -5x

This equation is similar to the standard equation of SHM

F = -kx, where k = constant.

So this is an SHM.

Now time period

= 2π√(m/k)

= 2π √{(0.2)/5}

= 2π √(0.04)

= 2π × 0.2

= 1.26 seconds.

Answer 2

Answer:

$\large\bold\red{(2)Yes,\:1.26\:sec.}$

Explanation:

Given,

A particle having,

• Potential Energy, U = $2.5{x}^{2}+100$ J

To find:

• If the motion is in S.H.M or not ?

We have to first find its force in terms of x.

Now,

We know that,

$\large\boxed{\bold{Force, F =-\frac{dU}{dx}}}$

Therefore,

We get,

$= > F = - \frac{d}{dx} (2.5 {x}^{2} + 100) \\ \\ = > F = - (2.5 \times 2x) \\ \\ = > \bold{F = - 5x}$

Now,

We know that,

For any particle having S.H.M,

The force must be in the form,

$\large\boxed{\bold{F=-kx\:;k \in constant}}$

Now,

Relating both the results,

We can clearly say that,

• The motion of particle is in S.H.M

Comparing the Coefficients,

We get,

• k = 5

Now,

We know that,

Time period of S.H.M is given by,

$\large \boxed{ \bold{T = 2\pi \sqrt{ \frac{m}{k} } }}$

Also,

It's given that,

• m = 0.2 Kg

Therefore,

Substituting the values,

We get,

$= > T = 2\pi \sqrt{ \frac{0.2}{5} } \\ \\ = > T = 2\pi \sqrt{ \frac{2}{50} } \\ \\ = > T = 2\pi \sqrt{ \frac{1}{25} } \\ \\ = > T = \frac{2\pi}{5} \\ \\ = > T = 0.4 \times 3.14 \\ \\ = > T = 1.256 \\ \\ = > \large \bold{T = 1.26 \: sec.}$

Hence,

The option (2) is the correct answer.