Question

force constant from shm equation

Answer 1

The force constant ($k$) in Simple Harmonic Motion (SHM) can be expressed as: $k = m\omega^2$ where $m$ is the mass and $\omega$ is the angular frequency.

Alternatively, in terms of linear frequency ($n$): $k = 4\pi^2 m n^2$

Answer 2

The force constant ($k$) in Simple Harmonic Motion (SHM) is derived from the fundamental equations governing the motion.

Explanation:

1. Defining Equation of SHM: The restoring force ($F$) in SHM is directly proportional to the displacement ($x$) from the equilibrium position and acts in the opposite direction. This is given by Hooke's Law: $F = -kx$ where $k$ is the force constant.

2. Newton's Second Law: According to Newton's second law, the force is also equal to mass ($m$) times acceleration ($a$): $F = ma = m\frac{d^2x}{dt^2}$

3. Equating the Forces: Combining the two expressions for force, we get the differential equation for SHM: $m\frac{d^2x}{dt^2} = -kx$ Rearranging this, we get: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

4. Comparing with Standard SHM Equation: The general differential equation for SHM is: $\frac{d^2x}{dt^2} + \omega^2x = 0$ where $\omega$ is the angular frequency of oscillation.

5. Relating Force Constant to Angular Frequency: By comparing the two differential equations, we can see that: $\omega^2 = \frac{k}{m}$ Therefore, the force constant $k$ can be expressed as: $k = m\omega^2$

6. Relating Angular Frequency to Linear Frequency: The angular frequency ($\omega$) is related to the linear frequency ($n$, the number of oscillations per second) by the relation: $\omega = 2\pi n$

7. Final Expression for Force Constant: Substituting the expression for $\omega$ into the equation for $k$: $k = m(2\pi n)^2$ $k = m(4\pi^2 n^2)$ $k = 4\pi^2 m n^2$

This equation gives the force constant in terms of the mass of the oscillating particle and its linear frequency.