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)=A0e−bt∕2m where A0 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)=‌

A0

2

when t=10s.
Setting Up the Equation:
We set ‌

A0

2

=A0e−10b∕2m
Divide both sides by A0 to get: ‌

1

2

=e−10b∕2m
Take the natural logarithm of both sides: ln(2)=‌

10b

2m

So, ‌

b

m

=‌

ln(2)

5

Now, How Long for Energy to Halve?
Mechanical energy E is proportional to the square of amplitude: E∝A2. It also decays exponentially, but at twice the rate: E(t)=E0e−bt∕m
We're looking for time when E(t)=‌

E0

2

So, ‌

E0

2

=E0e−bt∕m
Divide by E0:‌

1

2

=e−bt∕m
Take the natural log: ln(2)=‌

bt

m

Plug in the Value of fracbm:
From before, ‌

b

m

=‌

ln(2)

5

, so substitute this into the energy equation:
ln(2)=[‌

ln(2)

5

]t
Divide both sides by ‌

ln(2)

5

:t=5 seconds
Final Answer: It takes 5 seconds for the mechanical energy to become half of its original value.