When a mass of 28 g is attached to a certain

spring, it makes 19 complete vibrations in
3.9 s.
What is the spring constant of the spring?
Answer in units of N/m

a

To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

The formula for Hooke's Law is:

F = kx,

where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

In this case, we are given the mass that is attached to the spring (28 g = 0.028 kg) and the number of complete vibrations (19) that occur in a certain time period (3.9 s).

To find the spring constant, we need to calculate the period of one complete vibration, which is the time it takes for the mass to complete one full oscillation.

The period (T) can be calculated using the formula:

T = t / n,

where T is the period, t is the time taken for the given number of vibrations (3.9 s), and n is the number of vibrations (19).

Substituting the values, we get:

T = 3.9 s / 19 = 0.2053 s.

Now, we can calculate the angular frequency (ω) using the formula:

ω = 2π / T,

where ω is the angular frequency and T is the period.

Substituting the value of T, we have:

ω = 2π / 0.2053 s ≈ 30.631 rad/s.

Next, we can calculate the frequency (f) using the formula:

f = 1 / T,

where f is the frequency and T is the period.

Substituting the value of T, we have:

f = 1 / 0.2053 s ≈ 4.868 Hz.

Now, we can find the spring constant using the formula:

k = (2πf)²m,

where k is the spring constant, f is the frequency, and m is the mass.

Substituting the values, we get:

k = (2π(4.868 Hz))²(0.028 kg) ≈ 0.704 N/m.

Therefore, the spring constant of the spring is approximately 0.704 N/m.

The spring constant of the spring can be calculated using the formula:

k = (4π²m)/T²
where k is the spring constant in N/m, m is the mass in kg, and T is the time period in seconds.

First, we need to find the time period of one complete vibration:
T = (3.9 s) / (19 vibrations)
T = 0.2053 s/vibration

Now, we can use this time period and the given mass to find the spring constant:
k = (4π² * 0.028 kg) / (0.2053 s/vibration)²
k = 7.65 N/m

Therefore, the spring constant of the spring is 7.65 N/m.