Question

A body of mass $m$ is suspended from a spring of spring constant $k$. A damping force proportional to the velocity exerts itself on the mass. An appropriate representation of the motion is

• A. $m\frac{d^{2}x}{d{t}^2}-c\frac{dx}{dt}+kx=0$
• B. $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}-kx=0$
• C. $m\frac{d^{2}x}{d{t}^2}-c\frac{dx}{dt}-kx=0$
• D. $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$

Answer 1

The correct option is D $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$

The forces on the mass $m$ are due to spring and damper.

Suppose the mass moves by a distance $x$ from equilibrium and travels with a velocity $\frac{dx}{dt}$, then

Force due to spring is $-kx$ and force due to damper is $-c\frac{dx}{dt}$

By Newton's Second Law of motion, we have $m\frac{d^{2}x}{d{t}^2}=-kx-c\frac{dx}{dt}$

Thus, the equation of motion is $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$