
Consider one dimensional motion of a particle of mass . It has potential energy , where and are positive constants. At origin it has initial velocity . It performs simple harmonic oscillations. The frequency of the simple harmonic motion depends on:
A. and alone
B. , and alone
C. alone
D. and alone
Answer
500.4k+ views
Hint: Try to get the equation of acceleration using the second law of Newton mechanics to relate it to the acceleration of particles performing simple harmonic oscillation. For that, first, find the force by differentiating potential energy.
Formula used:
; Representing force(F) is the (negative)derivative of Potential energy( ).
, where is acceleration with respect to time, is angular velocity and is the displacement function.
Complete answer:
Here, a particle is performing a simple harmonic motion in one-dimensional motion which has potential energy at point from the mean position is .
We have,
Potential energy,
Where, and are constants
Now Force,
By putting the value of in the above equation, we get
Since the derivative of constant is zero, so
By substituting this, we have
(derivative of is )
We know that, from second law of newton where,
=mass of particle
=acceleration of particle
So,
Put the value of that we get in equation
The acceleration of a particle performing simple harmonic motion is given by,
Here, is the angular velocity of a particle.
By comparing and , we get
The frequency of simple harmonic motion depends on and .
So the correct option is A.
Note:
One interesting characteristic of the SHM of an object attached to a spring is the angular frequency, and therefore the period and frequency of the motion, depending on only the mass and the force constant, and not on other factors such as the amplitude of the motion.
Formula used:
Complete answer:
Here, a particle is performing a simple harmonic motion in one-dimensional motion which has potential energy at point
We have,
Potential energy,
Where,
Now Force,
By putting the value of
Since the derivative of constant is zero, so
By substituting this, we have
We know that,
So,
Put the value of
The acceleration of a particle performing simple harmonic motion is given by,
Here,
By comparing
So the correct option is A.
Note:
One interesting characteristic of the SHM of an object attached to a spring is the angular frequency, and therefore the period and frequency of the motion, depending on only the mass and the force constant, and not on other factors such as the amplitude of the motion.
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