Question

Compute the Inverse Laplace Transform of \(F(s) =\frac{s}{R^2s^2+16\pi^2}\)
take R=70

Answer 1

Step 1
Given that, R = 70.
The given function is, \(F(s) =\frac{s}{R^2s^2+16\pi^2}\)
\(F(s)=\frac{s}{70^2s^2+16\pi^2}\)
\(F(s)=\frac{s}{4900s^2+16\pi^2}\)
\(f(t)=L^{-1}\left(\frac{s}{4900s^2+16\pi^2}\right)\)
It is known that, \(L^{-1}(sF(s))=\frac{d}{(dt)}f(t)+f(0)\)
Step 2
Apply the above property and obtain the inverse Laplace transform of the given function as shown below.
\(f(t)=L^{-1}\left(\frac{s}{4900s^2+16\pi^2}\right)\)
\(f(t)=L^{-1}\left(\frac{1}{4900s^2+16\pi^2}\right)+L^{-1}\left(\frac{1}{4900s^2+16\pi^2}\right)(0)\)
\(f(t)=\frac{d}{(dt)}\left(\frac{1}{280} \sin \left(\frac{(2\pi t)}{35}\right)\right)+0\)
\(f(t)=\frac{1}{4900} \cdot \cos\left(\frac{2\pi t}{35}\right)\)
Thus, the required inverse Laplace transform is obtained.