A 1.3 g spider is dangling at the end of a silk thread. You can make the spider bounce up and down on the thread by tapping lightly on his feet with a pencil. You soon discover that you can give the spider the largest amplitude on his little bungee cord if you tap exactly once every 3 seconds.
What is the approximate spring constant of the silk thread?
Isn't f=1/2PI * sqrt(k/M)
f, of course, is 1/period
To find the approximate spring constant of the silk thread, we can use the formula for the period of a simple harmonic oscillator:
T = 2π√(m/k)
where T is the period, m is the mass, and k is the spring constant.
In this case, the period is given as 3 seconds, and the mass of the spider is 1.3 g (or 0.0013 kg). We can rearrange the formula to solve for the spring constant:
k = (4π²m) / T²
Substituting the known values:
k = (4π² * 0.0013) / (3²)
k ≈ 0.00228 N/m
Therefore, the approximate spring constant of the silk thread is approximately 0.00228 N/m.
To approximate the spring constant of the silk thread, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The equation for Hooke's Law is:
F = -kx
Where:
F is the force applied on the spring,
k is the spring constant,
x is the displacement from the equilibrium position.
In this case, the spider bouncing up and down on the silk thread can be considered as a simple harmonic motion (SHM) similar to a spring. The period (T) of the motion is given as 3 seconds.
The period of a simple harmonic motion is related to the spring constant by the formula:
T = 2π√(m/k)
Where:
T is the period,
π is a mathematical constant (approximately 3.14159),
m is the mass,
k is the spring constant.
We are given the period (T = 3 seconds) and the mass of the spider (m = 1.3 g = 0.0013 kg). We can rearrange the formula to solve for the spring constant (k):
k = (4π²m) / T²
Now, we can substitute the given values into the formula to calculate the spring constant:
k = (4π² * 0.0013) / (3²)
k = 5.74 N/m (approximately)
Therefore, the approximate spring constant of the silk thread is 5.74 N/m.