Answer:
τ = 5 s
Explanation:
When a vibrating body is damped. Its amplitude starts to decrease. This decrement is exponential. And it is given as follows:
[tex]X = X_{0}e^{-\frac{t}{2\tau}[/tex]
where,
τ = Time Constant = ?
X = Instantaneous value of amplitude
X₀ = Initial Value of amplitude
t = time interval = 10 s
The ratio of decrement is given as:
[tex]\frac{X}{X_0} = 36.8\% = 0.368[/tex]
therefore, using these values, we get:
[tex]\frac{X}{X_{0}} = 0.368 = e^{\frac{10\ s}{2\tau}}[/tex]
Taking natural log (ln) on both sides, we get:
[tex]ln(0.368) = \frac{10\ s}{2\tau}\\\\\tau = \frac{10\ s}{2ln(0.368)}[/tex]
τ = 5 s
The value of the time constant for the decrease in the amplitude of this oscillator is 5.
Given the following data:
To determine the value of the time constant:
Mathematically, the amplitude for damped harmonic motion is given by the formula:
[tex]X = X_o e^\frac{t}{2 \tau}[/tex]
Where:
Rearranging the formula, we have:
[tex]\frac{X}{X_o} = e^{-\frac{t}{2 \tau}}[/tex]
Substituting the given parameters into the formula, we have;
[tex]0.368 = e^\frac{-10}{2 \tau}\\\\ln(0.368) = \frac{-10}{2 \tau}\\\\-0.9997 = \frac{-10}{2 \tau}\\\\2 \tau \times -0.9997 = 10\\\\-1.9994\tau=-10\\\\\tau =\frac{-10}{-1.9994} \\\\\tau = 5.0[/tex]
Time constant = 5
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