A loudspeakers produce a musical sound by the oscillation of a diaphragm.if the amplitudes of oscillation is limited to i.20m,what the frequencies will result in the acceleration of the diaphragm exceeding g?

To determine the frequencies at which the acceleration of a loudspeaker diaphragm exceeds the acceleration due to gravity (g), we need to analyze the relationship between amplitude, frequency, and acceleration in oscillatory motion. Let's break this down step by step.

Understanding Oscillation and Acceleration

When a diaphragm in a loudspeaker oscillates, it moves back and forth in a periodic motion. The acceleration of this motion can be described using the formula:

a = -ω²x

In this equation:

• a is the acceleration of the diaphragm.
• ω (omega) is the angular frequency, which is related to the frequency (f) by the equation ω = 2πf.
• x is the displacement from the equilibrium position, which in this case is the amplitude of oscillation.

Setting Up the Problem

Given that the amplitude of oscillation is limited to 20 m, we want to find the frequencies that result in the acceleration exceeding the gravitational acceleration, which is approximately 9.81 m/s².

Calculating the Required Frequency

We can rearrange the acceleration formula to find the frequency:

9.81 < ω² * 20

Substituting ω with 2πf, we have:

9.81 < (2πf)² * 20

Now, simplifying this gives:

9.81 < 80π²f²

Next, we can isolate f²:

f² > 9.81 / (80π²)

Calculating the right side:

f² > 9.81 / (80 * 9.87) ≈ 0.0124

Taking the square root gives:

f > √0.0124 ≈ 0.1114 Hz

Conclusion on Frequency Values

This means that any frequency greater than approximately 0.1114 Hz will result in the acceleration of the diaphragm exceeding the acceleration due to gravity. In practical terms, this is a very low frequency, and in typical loudspeaker applications, frequencies are usually much higher, often in the range of tens to thousands of hertz.

To summarize, if the diaphragm oscillates with an amplitude of 20 m, it will produce accelerations exceeding g at frequencies above approximately 0.1114 Hz. This analysis highlights the relationship between amplitude, frequency, and acceleration in oscillatory systems, which is crucial for understanding how loudspeakers function.