Is the Motion Simple Harmonic? - Part 2

Classical Mechanics Level 3

This is a follow-up question for this problem. Solvers are encouraged to attempt it as well.

Consider the situation as shown in the diagram:

The point mass $m=1 kg$ , initially at the origin, is attached to two balls which are fixed to the points indicated on the diagram at all instants of time. The springs are identical with stiffness $k = 10 N/m$ and have an unstretched length of $L_o = 1 m$ . Gravity is absent in this scenario.

The mass is given an initial displacement of $x_o=0.01m$ to the right, along the x-axis and released. The goal is to compute the time period of the oscillatory motion of the mass. Let the time period be $T$ in seconds. Enter your answer as $\lfloor T \rfloor$ .

The answer is 234.

1 solution

A Former Brilliant Member
May 6, 2020

The acceleration of the particle is approximately given by

$v\dfrac{dv}{dx}=-\dfrac{k}{mL_0^2}x^3$ .

Integrating we get

$v=\dfrac{dx}{dt}=-\sqrt {\frac{k}{2mL_0^2}}\sqrt {a^4-x^4}$ , where $a=0.01=$ initial value of $x$ .

Integrating from the initial position to the equilibrium position ( $x=0$ ), we get

$\dfrac{1.311}{a}=\sqrt {\frac{k}{2mL_0^2}}\times \dfrac {T}{4}$ , where $T$ is the time period to be determined.

Substituting values, we get

$T=\dfrac{4\times 1.311}{0.01}\sqrt {\frac{2\times 1\times 1^2}{10}}\approx \boxed {234.5188}$ .

Hence the required answer is $\boxed {234}$ .

Glad to finally see a solution to this. Thank you for sharing

Karan Chatrath - 1 year, 1 month ago

Is the Motion Simple Harmonic? - Part 2

The answer is 234.

1 solution

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