1. A small fly of mass 0.26 g is caught in a spider's web. The web vibrates predominately with a frequency of 4.2 Hz. Find the effective spring stiffness constant k for the web. Hint: use natural frequency of oscillations formula.

N/m

2. At what frequency would you expect the web to vibrate if an insect of mass 0.52 g were trapped? Hint: use stiffness constant found in part 1. Hint: after the insect is trapped, the mass of the oscillator has increases by insect's mass, so the mass in the formula for natural frequency also has changed.

Hz

3. A spring scale being used to measure the weight of an object reads 12.8 N when it is used on earth. The spring stretches 4.95 cm under the load. The same object is weighed on the moon, where gravitational acceleration is
1/6g.

Find the reading of the spring scale on the moon.


Find the period for vertical oscillations of the spring on the moon.

(1) f = [1/(2 pi)]*(k/m)^1/2

Rearrange for k.

(2) With k staying the same, f drops a factor sqrt2 if the mass increases by a factor of 2.

(3) Spring scales measure weight, and weight = M g.

The scale reading decreases by a factor of 6. M stays the same and g decreases to 1/6 the value on Earth

1. To find the spring stiffness constant k for the spider's web, we can use the formula for the natural frequency of oscillations:

f = 1 / (2π√(m/k)),

where f is the frequency, m is the mass of the fly, and k is the stiffness constant. We have the frequency f = 4.2 Hz and the mass m = 0.26 g.

First, we need to convert the mass from grams to kilograms. Since 1 g = 0.001 kg, the mass of the fly is 0.26 g x 0.001 kg/g = 0.00026 kg.

Now, we can rearrange the formula to solve for k:

k = (1 / (2πf))^2 * m.

Plugging in the given values:

k = (1 / (2π * 4.2))^2 * 0.00026.

Calculating this expression will give us the value of k in N/m.

2. To find the frequency at which the web would vibrate if an insect of mass 0.52 g were trapped, we can use the same formula for the natural frequency of oscillations as in question 1:

f = 1 / (2π√(m/k)).

In this case, the mass m is the sum of the trapped insect's mass and the fly's mass: m = 0.52 g + 0.26 g.

Again, we need to convert the mass from grams to kilograms:

m = (0.52 g + 0.26 g) x 0.001 kg/g.

Once we have the new value for m, we can substitute it back into the formula and solve for f. This will give us the frequency in Hz.

3. To find the reading of the spring scale on the moon, we can use the formula:

F = mg,

where F is the weight of the object (force), m is the mass of the object, and g is the acceleration due to gravity.

Given that the weight on Earth is 12.8 N and the gravitational acceleration on the moon is 1/6g, we can rewrite the formula as:

F_moon = (m * (1/6g)),

where F_moon is the reading of the spring scale on the moon.

To find the period for vertical oscillations of the spring on the moon, we can use the formula:

T = 2π√(m/k),

where T is the period (time for one complete oscillation), m is the mass, and k is the spring stiffness constant.

We can use the same value for mass calculated in question 3 and the stiffness constant calculated in question 1 to find the period on the moon. The period will be in seconds.