Simple Harmonic Motion - When an insect gets caught in a spider web, its struggles cause the web to vibrate. This alerts the spider to a

Question

When an insect gets caught in a spider web, its struggles cause the web to vibrate. This alerts the spider to a potential meal. The frequency of vibration of the web gives the spider an indication of the mass of the insect. If we model the web as a spring:

(a) Would a rapidly vibrating web indicate a large (massive) or a small insect? Explain your reasoning.

(b) Suppose that 15-mg insect land on a horizontal web and depresses it 4.5 mm. What would be the web's effective spring constant?

(c) At what rate would the web-insect system vibrate, assuming that the web's mass is negligible compared to that of the insect?

(d) Would the vibration rate differ if the web were not horizontal?

Answer 1

@Diana: Since your comments indicate you want parts (b) and (c), here you go:

To compute a spring constant (part b), you need to know the distance that the spring is stretched or compressed due to a given force. Specifically:

(Force) = (Spring Constant) * (Distance)

The distance here is 4.5mm, and the force is the weight of the insect (which needs to be computed), using

Weight = (Mass)*(Acceleration due to Gravity)

It's fairly standard to express weight in Newtons (1 N = 1 kg*m/s^2), so you should convert the mass to kg first:

15 mg * (1 g / 1000 mg) * (1 kg / 1000 g) = 1.5 * 10^(-5) kg.

Using acceleration due to gravity at 10 m/s^2 (use 9.8 if you want better accuracy), the weight is thus 1.5*10^(-4) N.

Also convert distance to m:

4.5 mm * (1 m / 1000 mm) = 0.0045 m

Thus the Spring Constant = Weight / Distance = 0.0333 kg/s^2

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For (c), the frequency (rate) of vibration for simple harmonic motion is 1/(2pi) * sqrt(k/m), so just insert the values from above.