Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

Consider one dimensional motion of a particle of massm. It has potential energy U=a+bx2, where a and b are positive constants. At origin (x=0) it has initial velocity v0. It performs simple harmonic oscillations. The frequency of the simple harmonic motion depends on:
A. b and m alone
B. b, a and m alone
C. b alone
D. b and a alone

Answer
VerifiedVerified
500.4k+ views
like imagedislike image
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:
 F=dUdx; Representing force(F) is the (negative)derivative of Potential energy(U).
a(t)=ω2x(t), where a(t) is acceleration with respect to time, ω is angular velocity and x(t) 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 x from the mean position is U.
We have,
Potential energy, U=a+bx2
Where, a and b are constants
Now Force, F=dUdx
 By putting the value of Uin the above equation, we get
F=d(a+bx2)dxF=[ddx(a)+ddx(bx2)]
Since the derivative of constant is zero, so
ddx(a)=0
By substituting this, we have
F=0bddx(x2)
F=0b2x (derivative of xn is nx(n1))
F=2bx(1)
We know that, F=ma from second law of newton where,
m=mass of particle
a=acceleration of particle
So,
a=Fm
Put the value of Fthat we get in equation (1)
a=Fm=2bmx(2)
The acceleration of a particle performing simple harmonic motion is given by,
a(t)=ω2x(t)(3)
Here, ω is the angular velocity of a particle.
By comparing (2)and(3), we get
ω2=2bmω=2bm

The frequency of simple harmonic motion depends on band m.
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.