A 0.461 kg mass is attached to a spring and executes simple harmonic motion with a pe- riod of 0.59 s. The total energy of the system is 2.4 J.
Find the force constant of the spring
To find the force constant (k) of the spring, we can use the formula for the period (T) of a mass-spring system:
T = 2π√(m/k)
Where:
T = Period of motion
m = Mass of the object
k = Force constant of the spring
In this case, we are given the mass (m) and the period (T). We can rearrange the formula to solve for the force constant (k):
k = (4π²m) / T²
Let's plug the values into the formula:
m = 0.461 kg
T = 0.59 s
k = (4π² × 0.461) / (0.59)²
Now we can calculate the force constant (k) using a calculator:
k ≈ 20.85 N/m
Therefore, the force constant of the spring is approximately 20.85 N/m.
To find the force constant of the spring, we can use the formula for the total energy of a simple harmonic motion:
Total Energy = (1/2) * k * A^2
Where:
k is the force constant of the spring
A is the amplitude of the motion
We are given the total energy (2.4 J) and the mass (0.461 kg) of the attached mass. We need to find the force constant (k).
First, let's find the amplitude of the motion. The period of the motion (T) is given as 0.59 seconds. The formula for the period of simple harmonic motion is:
T = (2 * pi) * sqrt(m / k)
Where:
T is the period
m is the mass of the object
k is the force constant of the spring
Rearranging this equation, we can solve for k:
k = (4 * pi^2 * m) / T^2
Plugging in the values, we have:
k = (4 * pi^2 * 0.461 kg) / (0.59 s)^2
k ≈ 68.387 N/m
Therefore, the force constant of the spring is approximately 68.387 N/m.