When a 0.213-kg mass is attached to a vertical spring, it causes the spring to stretch a distance d. If the mass is now displaced slightly from equilibrium, it is found to make 150 oscillations in 52.0 s.
Find the stretch distance, d.
To find the stretch distance, d, we need to first calculate the spring constant, k, using Hooke's Law, and then use the formula for the period of a mass-spring system to find the period, T. Finally, we can use the relationship between the period and the number of oscillations to find the stretch distance, d.
Here's how we can approach this problem step by step:
Step 1: Calculate the spring constant, k.
According to Hooke's Law, the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Therefore, we can write the equation as:
F = kx
Where F is the force applied by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this problem, the mass, m, is 0.213 kg, and the force due to gravity, Fg, is given by:
Fg = mg
Where g is the acceleration due to gravity (approximated as 9.8 m/s^2).
Since the equilibrium position is when the spring is unstretched, the force exerted by the spring, Fs, is equal to the force due to gravity at equilibrium:
Fs = Fg
Therefore, we can write the equation as:
kx = mg
Now we can solve for the spring constant, k:
k = mg / x
Step 2: Calculate the period, T.
The period of a mass-spring system is the time it takes for one complete oscillation.
The formula for the period, T, of a mass-spring system is given by:
T = 2π√(m/k)
Here, m is the mass (0.213 kg), and k is the spring constant that we calculated in step 1.
Step 3: Calculate the stretch distance, d.
The relationship between the period and the number of oscillations, N, is given by:
T = t / N
Where t is the total time for N oscillations.
In this problem, we are given that the number of oscillations, N, is 150, and the total time, t, is 52.0 s.
Rearranging the equation, we can find the period:
T = t / N
Now, we can substitute the values and calculate the period, T.
Finally, we can find the stretch distance, d, by rearranging the formula for the period of a mass-spring system:
T = 2π√(m/k)
Rearranging the equation, we get:
d = (T^2 * k) / (4π^2)
Now, substitute the values we've calculated for T and k to find the stretch distance, d.