A spring stretches 3.4-cm when a 7-g object is hung from it. The object is replaced with a block of mass 24 g that oscillates in simple harmonic motion. Calculate the period of motion
To calculate the period of motion, we need to know the force constant of the spring (also known as the spring constant) and the mass of the block. The force constant is a measure of how stiff the spring is, while the mass of the block determines the inertia of the system.
1. First, let's calculate the force constant of the spring using Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be written as F = kx, where F is the force applied, k is the force constant, and x is the displacement.
Given that the spring stretches 3.4 cm (or 0.034 m) when a 7 g object is hung from it, we can find the force constant using the equation F = mg, where m is the mass and g is the acceleration due to gravity.
F = mg
kx = mg
k = mg/x
Here, m = 7 g = 0.007 kg and x = 0.034 m.
k = (0.007 kg)(9.8 m/s²) / 0.034 m
k ≈ 2.03 N/m
Therefore, the force constant of the spring is approximately 2.03 N/m.
2. Next, we can calculate the period of motion using the formula for the period of simple harmonic motion, T = 2π√(m/k), where T is the period, m is the mass, and k is the force constant.
Given that the mass is 24 g = 0.024 kg, and we already found the force constant to be 2.03 N/m, we can substitute these values into the formula:
T = 2π√(m/k)
T = 2π√(0.024 kg / 2.03 N/m)
Evaluating this expression gives us the period of motion.
T ≈ 2.22 s
Therefore, the period of motion for the block is approximately 2.22 seconds.