When a mass of 28 g is attached to a certain
spring, it makes 18 complete vibrations in
4.1 s.
What is the spring constant of the spring?
Answer in units of N/m.
To calculate 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.
Hooke's Law equation: F = -kx
Where:
- F is the force applied to the spring (in Newtons, N)
- k is the spring constant (in N/m)
- x is the displacement of the spring from its equilibrium position (in meters)
In this case, we can find the spring constant by dividing the force by the displacement.
Since the problem provides information about the number of vibrations in a given time, we need to calculate the displacement first.
The number of complete vibrations is equal to the number of cycles the spring completes. One complete vibration is equal to the displacement changing from the maximum amplitude on one side to the maximum amplitude on the other side.
In this case, since the problem mentions 18 complete vibrations, it implies that the displacement is equal to 18 times the amplitude of the vibration.
We also have the time it takes for the vibrations, which is given as 4.1 seconds. We need to convert this to the time period of one complete vibration.
Time period (T) is the time it takes for one complete vibration. It can be calculated by dividing the total time (4.1 seconds) by the number of complete vibrations (18).
T = total time / number of complete vibrations
T = 4.1 s / 18 = 0.2278 s
Now that we have the time period of one complete vibration, we can calculate the frequency (f). Frequency is the reciprocal of time period.
f = 1 / T = 1 / 0.2278 s ≈ 4.387 Hz
Now, let's calculate the displacement (x) by multiplying the frequency (f) by the time period (T).
x = f * T ≈ 4.387 Hz * 0.2278 s ≈ 1.0007 m
Finally, we can calculate the spring constant (k) by dividing the force by the displacement.
The force (F) can be calculated using the mass (m) attached to the spring and the acceleration due to gravity (g). The force exerted by the mass is given by F = m * g.
In this case, the mass is given as 28 g. To use SI units, we need to convert grams to kilograms by dividing by 1000 (1 kg = 1000 g). Therefore, the mass is 0.028 kg.
The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Now, we can calculate the force:
F = m * g = 0.028 kg * 9.8 m/s^2 ≈ 0.2744 N
Finally, we can calculate the spring constant (k) using Hooke's Law equation:
k = F / x ≈ 0.2744 N / 1.0007 m ≈ 0.274 N/m
Therefore, the spring constant of the spring is approximately 0.274 N/m.