How do you find time period of an oscillatory motion(not SHM)

[jd12345]

Homework Statement

Force ( F ) = -kx /√( x^2 + r^2 ) where k is a constant and r(constant-amplitude) is the distance initially the object is away from the origin, x(variable) is distance of the object from the origin. Find time period of this oscillation




2. The attempt at a solution
Clearly Force is not proportional to distance so its not an SHM. For an SHM its easy to find time period - there is a simple formula but i have no idea how to do it for other oscillatory motions. I wrote acceleration as the second derivative of distance and then tried to integrate but its too lengthy and i got stuck in the integral too
IS there an easy method to find the time period of an oscillatory motion which is not an SHM
IS integrating the only way - if it is i'll have to post it in calculus section

 

[anathema]

Thank you for your question. I am a scientist and I would be happy to assist you in finding the time period of this oscillation.

Firstly, I would like to clarify that the equation given for force is actually a harmonic oscillator equation, which is a type of simple harmonic motion (SHM). This type of oscillation occurs when the force acting on an object is directly proportional to the displacement from equilibrium.

To find the time period of this oscillation, we can start by writing the equation of motion for the system:

ma = -kx /√( x^2 + r^2 )

Where m is the mass of the object and a is the acceleration. We can rewrite this equation as:

a + (k/m) * x /√( x^2 + r^2 ) = 0

This is now in the form of a standard harmonic oscillator equation, where ω^2 = k/m. Therefore, the time period for this oscillation can be found using the formula:

T = 2π/ω = 2π√(m/k)

To find the time period, we need to know the values of m and k. If these values are not given in the problem, we can use the initial conditions of the system to determine them. For example, if we know the amplitude and the initial distance of the object from the origin, we can use the equation for potential energy to find k, and then use the equation for kinetic energy to find m.

Alternatively, we can also use the given equation for force to find k and then use the equation for potential energy to find m. Once we have these values, we can plug them into the formula for time period to get the final answer.

I hope this explanation helps you in finding the time period of this oscillation. If you have any further questions, please feel free to ask. Thank you for your interest in science.