Solving Wave Motion Problem: Finding Amplitude for Objects to Leave Ground

[Klinger]

I've looked at the following problem several times and have gotten stuck.

The Problem
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An earthquake-produced suface wave can be approximated by a sinusoidal transverse wave. Assuming a frequency of 0.50 Hz (typical of of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact with the ground?

My Thoughts So Far
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To leave the ground acceleration must be 9.8 m/s^2. The equation of motion is x = A cos 2 Pi f t, where A is amplitude, f is frequency and t is time. I'm not sure how go from this point to get the equation for acceleration.

What's the next step?

 

[Gokul43201]

Find the second derivative of x with respect to t, and that's your acceleration.

Then, you want to find the maximum value that this can take, which is easy, since trigonometric functions are bounded.

 

[wais]

First of all, it's great that you have attempted to solve the problem and have identified the key components such as frequency and acceleration. To find the amplitude needed for objects to leave the ground, we can use the equation for acceleration: a = -ω^2x, where ω is the angular frequency (2πf).

Next, we can plug in the given values for frequency and acceleration and solve for amplitude:

9.8 m/s^2 = - (2π(0.50 Hz))^2 A

A = 9.8 m/s^2 / (4π^2 * 0.25 Hz^2)

A = 9.8 / 3.93 ≈ 2.49 m

Therefore, an amplitude of 2.49 meters is needed for objects to leave the ground when subjected to a 0.50 Hz earthquake-produced surface wave.

I hope this helps and feel free to ask for clarification if needed. Keep up the problem-solving mindset!