A damped oscillatory motion can be modeled by combining an exponential decay function with a sine function. In this context, consider a function of the form \(f(x) = Ae^{-kx}\sin(bx+c)\). Use the information provided to:
(a) Form a combined model that represents a damped oscillation where the amplitude decays by half every unit time, and the sine function completes one full cycle in a period of \(2\pi\). Assume appropriate values for \(A\), \(k\), \(b\), and \(c\) to illustrate this behavior.
(b) Interpret the model for HL students by explaining the roles of the exponential and sine components in representing the physical phenomenon. Clearly discuss the significance of the parameters \(k\) and \(b\).
Question
Form a combined model for the damped oscillation by assigning appropriate values. Hint: choose \(A = 1\), set the decay constant \(k\) such that the amplitude halves when \(x = 1\) (i.e., \(e^{-k}=\frac{1}{2}\)), choose \(b = 1\) to complete one cycle in \(2\pi\) units, and let \(c = 0\). Write down the explicit form of \(f(x)\).
Hint
Hint
Hint
Use the relationship \(e^{-k} = \frac{1}{2}\) to find \(k\). Remember that \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
- Formula Booklet
Complete Answer
\(f(x) = e^{-\ln(2)x}\sin(x)\)
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Interpret the model provided in part (a) by explaining how the exponential component and the sine component contribute to representing a damped oscillatory behavior. Clearly discuss the significance of the parameters \(k\) and \(b\).
Hint
Hint
Hint
Discuss how \(e^{-\ln(2)x}\) represents a halving of amplitude and \(\sin(x)\) represents periodic oscillations. Explain how \(k\) controls the rate of damping and \(b\) controls the frequency of oscillation.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
- Formula Booklet
Complete Answer
The exponential term \(e^{-\ln(2)x}\) models the damping effect, causing the amplitude to halve with each unit increase in \(x\). The parameter \(k = \ln(2)\) dictates the rate of this decay. The sine term \(\sin(x)\) provides the periodic oscillations with a period of \(2\pi\), and the parameter \(b = 1\) determines this frequency of oscillation.
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
