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)= when
t=10s.
Setting Up the Equation:
We set
=A0e−10b∕2mDivide both sides by
A0 to get:
=e−10b∕2mTake the natural logarithm of both sides:
ln(2)=So,
=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∕mWe're looking for time when
E(t)=So,
=E0e−bt∕mDivide by
E0:=e−bt∕mTake the natural log:
ln(2)=Plug in the Value of fracbm:
From before,
=, so substitute this into the energy equation:
ln(2)=[]tDivide both sides by
:t=5 seconds
Final Answer: It takes 5 seconds for the mechanical energy to become half of its original value.
