The amplitude of a damped oscillator decreases to 0.9 times its original magnitude is 5s. In another 10s it will decrease to a times its original magnitude, where a equals.

To determine the value of \( a \) for the damped oscillator, we need to understand how the amplitude of a damped oscillator behaves over time. The damping process typically follows an exponential decay model. Let's break it down step by step.

Understanding Damped Oscillation

A damped oscillator experiences a gradual reduction in amplitude due to energy loss, often from friction or resistance. The amplitude \( A(t) \) at any time \( t \) can be expressed with the formula:

A(t) = A_0 e^{-\beta t}

Here, \( A_0 \) is the initial amplitude, \( \beta \) is the damping coefficient, and \( e \) is Euler's number (approximately 2.71828). The term \( e^{-\beta t} \) represents the decay of amplitude over time.

Amplitude Reduction Over Time

From your question, we know that the amplitude decreases to 0.9 times its original value in 5 seconds. We can set up the following equation:

0.9A_0 = A_0 e^{-\beta \cdot 5}

Dividing both sides by \( A_0 \) (assuming \( A_0 \) is not zero), we simplify to:

0.9 = e^{-\beta \cdot 5}

Taking the natural logarithm of both sides gives us:

\ln(0.9) = -\beta \cdot 5

From this, we can solve for \( \beta \):

\beta = -\frac{\ln(0.9)}{5}

Calculating Further Damping

Now, we need to find the amplitude after another 10 seconds, which means a total of 15 seconds. We can use the same exponential decay formula:

A(15) = A_0 e^{-\beta \cdot 15}

Substituting \( \beta \) from our earlier calculation, we have:

A(15) = A_0 e^{-(-\frac{\ln(0.9)}{5}) \cdot 15}

This simplifies to:

A(15) = A_0 e^{-\ln(0.9) \cdot 3} = A_0 (0.9)^3

Since \( (0.9)^3 \) equals \( 0.729 \), we find that:

A(15) = 0.729 A_0

Final Result

Thus, after a total of 15 seconds, the amplitude decreases to approximately 0.729 times its original magnitude. Therefore, the value of \( a \) is:

a = 0.729

In summary, the damped oscillator's amplitude decreases progressively over time, and in this case, it reduces to about 72.9% of its original magnitude after 15 seconds. This behavior highlights the impact of damping on oscillatory systems and how we can mathematically describe these changes.