when a .187 kg mass is attached to a vertical spring it causes the spring to stretch distance d. if the mass is now displaced slightly from equilibrium, it is found to make 223 oscillations in 157s. Find the stretch distance d.
To find the stretch distance (d) of the spring, we can use the formula for the period of oscillation (T) of a mass-spring system. The period of oscillation is the time taken for one complete cycle of oscillation.
The formula for the period of oscillation is given by:
T = 2π√(m/k)
Where:
T = Period of oscillation
m = Mass attached to the spring
k = Spring constant
In this case, we are given the mass (m) and the number of oscillations (n) made in a certain time (t). We can find the period of oscillation using this information.
First, find the period of one oscillation using the formula:
T = t / n
T = 157s / 223
T ≈ 0.7036s
Now, we can rearrange the formula for the period of oscillation to solve for the spring constant (k):
k = (4π²m) / T²
k = (4π² * 0.187 kg) / (0.7036s)²
k ≈ 16.642 N/m
Finally, we can use the spring constant and the mass to find the stretch distance (d) using Hooke's Law:
F = k * d
Where:
F = Force applied to the spring
The force applied to the spring is the weight of the mass:
F = m * g
Where:
g = Acceleration due to gravity (approximately 9.8 m/s²)
F = 0.187 kg * 9.8 m/s²
F ≈ 1.8356 N
Now we can find the stretch distance (d) using Hooke's Law:
d = F / k
d ≈ 1.8356 N / 16.642 N/m
d ≈ 0.1103 meters (0.1103 m)