Question 6.14: Small oscillations in constrained motion A particle P of mas...
Small oscillations in constrained motion
A particle P of mass m can slide freely along a long straight wire. P is connected to a fixed point A, which is at a distance 4a from the wire, by a light elastic cord of natural length 3a and strength α. Find the approximate period of small oscillations of P about its equilibrium position.
Suppose P has displacement x from its equilibrium position. In this position, the length of the cord is \left(16 a^{2}+x^{2}\right)^{1 / 2} and its potential energy V is
\begin{aligned}V &=\frac{1}{2} \alpha\left[\left(16 a^{2}+x^{2}\right)^{1 / 2}-3 a\right]^{2} \\&=\frac{1}{2} \alpha\left[25 a^{2}+x^{2}-6 a\left(16 a^{2}+x^{2}\right)^{1 / 2}\right]\end{aligned}
The energy conservation equation for P is therefore
\frac{1}{2} m \dot{x}^{2}+\frac{1}{2} \alpha\left[25 a^{2}+x^{2}-6 a\left(16 a^{2}+x^{2}\right)^{1 / 2}\right]=E,
which, on neglecting powers of x higher than the second, becomes
\frac{1}{2} m \dot{x}^{2}+\frac{1}{2} \alpha\left[a^{2}+\frac{x^{2}}{4}\right]=E.
On differentiating this equation with respect to t, we obtain the approximate linearised equation of motion
m \ddot{x}+\frac{\alpha}{4} x=0.
This is the SHM equation with ω² = α/4m. It follows that the approximate period of small oscillations about x = 0 is 4 \pi(m / \alpha)^{1 / 2} .