A spring mass oscillator consists of a spring of constant 2500 N/m and a mass 1 kg. The amplitude of the oscillator's motion is 0.17 meters. What...

Wednesday, August 31, 2016

A spring mass oscillator consists of a spring of constant 2500 N/m and a mass 1 kg. The amplitude of the oscillator's motion is 0.17 meters. What...

A spring-mass oscillator will undergo simple harmonic motion (SHM). Such motion can be explained by the following equations:

$\displaystyle{x}={A}{\cos{{\left(\omega{t}+\phi\right)}}}$

$\displaystyle{v}=-\omega{A}{\sin{{\left(\omega{t}+\phi\right)}}}$

where, x (displacement from the equilibrium position) = 0.136 m,

A (amplitude) = 0.17 m

and, $\displaystyle\omega=\sqrt{{\frac{{k}}{{m}}}}=\sqrt{{\frac{{2500}}{{1}}}}={50}\frac{{{r}{a}{d}}}{{s}}$

substituting the values of various parameters in the equation of displacement, we get:

$\displaystyle{0.136}={0.17}{\cos{{\left(\omega{t}+\phi\right)}}}$

solving this equation, we get, $\displaystyle\omega{t}+\phi={0.644}{r}{a}{d}{i}{a}{n}{s}$

or, $\displaystyle\omega{t}+\phi={5.64}{r}{a}{d}{i}{a}{n}{s}$

substituting this value in the equation of velocity, we get:

$\displaystyle{v}=-{50}\times{0.17}{\sin{{\left({0.644}\right)}}}$

solving this equation, we get, v = - 5.1 m/s.

Another solution for velocity, v = $\displaystyle-{50}\times{0.17}{\sin{{\left({5.64}\right)}}}$

or, v = 5.1 m/s.

Speed = magnitude of velocity = 5.1 m/s

Thus, the mass of 1 kg is moving at a speed of 5.1 m/s when it is at a distance of 0.136 m from the equilibrium position. Note that mass can have a velocity of either 5.1 m/s or -5.1 m/s at the given displacement, depending on the direction of motion (towards or away from the equilibrium). However, speed is only a scalar quantity and does not have a direction. Hence the speed of the mass at that particular displacement (x = 0.136 m) is 5.1 m/s.

Hope this helps.

Notes

Wednesday, August 31, 2016