TG EAMCET 3 May 2025 Shift 1 Paper

Understanding the Amplitude Change:The formula for how the amplitude (A) of a damped oscillator changes over time is: $A(t)=A_0 e^{-\frac{b t}{2 m}}$ where $A_0$ is the starting amplitude, $b$ is the damping constant, $m$ is the mass, and $t$ is time.We are told the amplitude drops to half its original value in 10 seconds. So, $A(t)=\frac{A_0}{2}$ when $t=10\,\text{s}$.Setting Up the Equation:We set $\frac{A_0}{2}=A_0 e^{-\frac{10 b}{2 m}}$Divide both sides by $A_0$ to get: $\frac{1}{2}=e^{-\frac{10 b}{2 m}}$Take the natural logarithm of both sides: $\ln(2)=\frac{10 b}{2 m}$So, $\frac{b}{m}=\frac{\ln(2)}{5}$Now, How Long for Energy to Halve?Mechanical energy $E$ is proportional to the square of amplitude: $E \propto A^2$. It also decays exponentially, but at twice the rate: $E(t)=E_0 e^{-\frac{b t}{m}}$We're looking for time when $E(t)=\frac{E_0}{2}$So, $\frac{E_0}{2}=E_0 e^{-\frac{b t}{m}}$Divide by $E_0: \frac{1}{2}=e^{-\frac{b t}{m}}$Take the natural log: $\ln(2)=\frac{b t}{m}$Plug in the Value of fracbm:From before, $\frac{b}{m}=\frac{\ln(2)}{5}$, so substitute this into the energy equation:$\ln(2)=\left[\frac{\ln(2)}{5}\right] t$Divide both sides by $\frac{\ln(2)}{5}: t=5$ secondsFinal Answer: It takes 5 seconds for the mechanical energy to become half of its original value.