Question

A spring-mass oscillator hanging from the ceiling is damped by a vane immersed in a liquid. The mass is 250 g, k is 85 N/m and a damping constant of 70 g/s.

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a. Sketch the setup above. [2] b. Derive the expression of the mechanical energy for under damped system. [10] c. Calculate the period of the motion. [2] d. How long does it take for the amplitude of the damped oscillations to drop to half its initial value? [4] e. Find the time it will take for the mechanical energy to drop to half its initial value.

Answer 1

Given that,

Mass of spring = 250 g

Spring constant = 85 N/m

Damping constant = 70 g/s = 70/1000 kg/s

(a) Sketch the setup.

(b). We need to calculate the expression of the mechanical energy for under damped system

Using formula of mechanical energy

$E=\dfrac{1}{2}kx^2+\dfrac{1}{2}m\dot{x^2}$

(c). We need to calculate the period of motion

Using formula of period

$T=2\pi\sqrt{\dfrac{m}{K}}$

Put the value into the formula

$T=2\pi\sqrt{\dfrac{0.25}{85}}$

$T=0.34\ sec$

(d). Amplitude at time t is $x_{m}e^{\dfrac{-bt}{2m}}$

The amplitude has value $x_{m}$ at t =0,

We need to calculate the value of  t

Using formula of amplitude

$x_{m}e^{\dfrac{-bt}{2m}}=\dfrac{1}{2}x_{m}$

$e^{\dfrac{-bt}{2m}}=\dfrac{1}{2}$

Taking log on both side

$ln(e^{\dfrac{-bt}{2m}})=ln(\dfrac{1}{2})$

$t=\dfrac{-2m ln(\dfrac{1}{2})}{b}$

Put the value into the formula

$t=\dfrac{-2\times(0.25)\times ln(\dfrac{1}{2})}{0.070}$

$t=4.9\ s$

(e). We need to calculate the time it will take for the mechanical energy to drop to half its initial value

Using formula of energy

$E(t)=\dfrac{1}{2}kx_{m}^2e^{\dfrac{-bt}{m}}$

It has value $\dfrac{1}{2}kx_{m}^2$ at t = 0

At time t,

$\dfrac{1}{2}Kx_{m}^2e^{\dfrac{-bt}{m}}=\dfrac{1}{2}(\dfrac{1}{2}kx_{m}^2)$

$e^{\dfrac{-bt}{m}}=\dfrac{1}{2}$

Taking log on both side

$t=\dfrac{-m ln(\dfrac{1}{2})}{b}$

Put the value into the formula

$t=\dfrac{-(0.25)\times ln(\dfrac{1}{2})}{0.070}$

$t=2.5\ sec$

Hence, (b). The expression of the mechanical energy for under damped system is $E=\dfrac{1}{2}kx^2+\dfrac{1}{2}m\dot{x^2}$

(c). The period of motion is 0.34 sec.

(d). The value of t is 4.9 sec.

(e). The value of time is 2.5 sec.