A loudspeaker has a movable diaphragm (cone) that vibrates back and forth in harmonic motion to produce sound. The displacement of the cone playing a sinusoidal test tone is shown above. Find the following values from the graph.
(NOTE: t = 0.03 sec, and the vertical axis of the graph goes from -0.02 m to 0.02 m).
the amplitude A = m
the period T = sec
the frequency f = Hz
the angular frequency ω = rad/sec
max displacement xmax = m
max speed vmax = m/s
max acceleration amax = m/s2
To find the values from the graph, we need to analyze the given information.
1. Amplitude (A):
The amplitude is the maximum displacement of the cone from its equilibrium position. In this case, we can see from the graph that the displacement reaches a maximum value of 0.02 m. So, the amplitude (A) is 0.02 m.
2. Period (T):
The period is the time taken for one complete cycle of the harmonic motion. From the graph, we can observe that one complete cycle takes approximately 0.06 seconds (from one peak to the next peak). Therefore, the period (T) is 0.06 s.
3. Frequency (f):
The frequency is the number of cycles per second. It is the reciprocal of the period. So, the frequency (f) can be calculated by taking the reciprocal of the period. In this case, f = 1 / T = 1 / 0.06 s = 16.67 Hz.
4. Angular frequency (ω):
The angular frequency is related to the frequency by the formula ω = 2πf, where π is a constant (approximately 3.14159). Substituting the value of f into the formula, we get ω = 2π * 16.67 Hz = 104.72 rad/s.
5. Maximum displacement (xmax):
From the graph, we can see that the maximum displacement of the cone is 0.02 m (same as the amplitude). So, xmax = 0.02 m.
6. Maximum speed (vmax):
The maximum speed occurs when the displacement is zero (at the equilibrium position) and the slope of the graph is maximum. From the given graph, we can observe that between t = 0.02 s to t = 0.04 s, the displacement is zero and the slope is steepest. Therefore, between these two points, the cone achieves its maximum speed. However, without specific values or precise measurements, we cannot determine the exact numeric value of vmax.
7. Maximum acceleration (amax):
The maximum acceleration occurs when the displacement is at its maximum (amplitude) and the slope is zero. From the graph, we can observe that at t = 0.03 s, the displacement is at its maximum and the slope is zero. Therefore, at t = 0.03 s, the cone experiences its maximum acceleration. However, without specific values or precise measurements, we cannot determine the exact numeric value of amax.