Damped Oscillations
1. (a) For each of following graphs, determine
if graph shows damped oscillation.
(b) if it shows damped oscillation :
(i) identify damping factor
(ii)where damping occurs
Note : use a graphing calculator to sketch these graphs
f(x) = ¡î5 cos 1.2 x
f(x) = x sin 4x
f(x) = x©ø sin 5x
f(x)=squareroot5cos1.2x
f(x)=xsin4x
f(x)=x^3sin5x
is it no damping for all of them
None of these equation have damping
To determine if a graph represents damped oscillation, you need to analyze the behavior of the graph with respect to time. Here's how you can approach this problem using a graphing calculator:
(a) For each of the following graphs, determine if the graph shows damped oscillation:
(i) f(x) = -5cos(1.2x)
- Enter the function into the graphing calculator.
- Observe the graph. If the graph shows a repetitive pattern with a decaying amplitude over time, it indicates damped oscillation.
(ii) f(x) = xsin(4x)
- Enter the function into the graphing calculator.
- Observe the graph. If the graph shows a repetitive pattern with a decaying amplitude over time, it indicates damped oscillation.
(iii) f(x) = x^(2/3)sin(5x)
- Enter the function into the graphing calculator.
- Observe the graph. If the graph shows a repetitive pattern with a decaying amplitude over time, it indicates damped oscillation.
(b) If the graph shows damped oscillation, perform the following steps:
(i) Identify the damping factor:
- The damping factor is a measure of how fast the amplitude of the oscillation decays over time.
- Look for the rate at which the amplitude decreases. The damping factor can be estimated by observing the rate of decay in the amplitude of the oscillation.
(ii) Identify where damping occurs:
- The damping can occur in different regions of the graph, depending on the specific function.
- Look for areas where the amplitude decreases rapidly. These regions indicate where damping occurs.
By following these steps with a graphing calculator, you can determine if a graph represents damped oscillation, identify the damping factor, and locate where damping occurs on the graph.