A mass m is gently placed on the end of a freely hanging spring. The mass then falls 21.3 cm before it stops and begins to rise. What is the frequency of the oscillation?
I found the same question (with a different length) posted on here, but following the steps of the answer provided is not giving me a correct answer. I have used every resource I can think of, and have tried this problem over and over, but I cannot seem to get the correct answer. Any help would be great!!
To find the frequency of the oscillation, you need to use Hooke's Law and the formula for the period of a mass-spring system.
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as:
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.
When the mass is gently placed on the spring, it will stretch the spring and create a displacement. Using Hooke's Law, we can write:
-mg = -kx
where m is the mass, g is the acceleration due to gravity, and x is the displacement. The negative signs indicate that the force and displacement are in opposite directions.
Now, let's solve for the spring constant k:
k = mg / x
The period T of a mass-spring system is the time taken for one complete oscillation. It can be calculated using the formula:
T = 2π√(m/k)
Substituting the value of k, we get:
T = 2π√(m / (mg / x))
= 2π√(x / g)
The frequency of oscillation (f) is the reciprocal of the period:
f = 1 / T
= 1 / (2π√(x / g))
Now, let's plug in the given information to find the frequency. You mentioned that the mass falls 21.3 cm, which means the displacement (x) is 21.3 cm or 0.213 m. The acceleration due to gravity (g) can be approximated as 9.8 m/s².
f = 1 / (2π√(0.213 / 9.8))
≈ 1 / (2π√(0.0218))
≈ 1 / (2π * 0.0467)
≈ 1 / 0.2926
≈ 3.41 Hz
Therefore, the frequency of the oscillation is approximately 3.41 Hz. If you're not obtaining the same answer, double-check your calculations and make sure you're using the correct values for the displacement (x) and acceleration due to gravity (g).