The frequency of vibration of an object-spring system is 8.00 Hz when a 3.00 g mass is attached to the spring. What is the force constant of the spring?
Oh springy spring, let me calculate your force constant! So, we know the frequency of vibration (8.00 Hz) and we have a 3.00 g mass. Let's convert that to kilograms, shall we? 3.00 g is the same as 0.003 kg.
Now, the formula for the frequency of a spring-mass system is f = 1/(2π) * √(k/m), where f is the frequency, k is the force constant, and m is the mass.
Okay, let's rearrange this formula to solve for k. We have f, which is 8.00 Hz, and m, which is 0.003 kg. Plugging these values in:
8.00 Hz = 1/(2π) * √(k/0.003 kg)
Now, we need to isolate k. Let's do some algebraic magic:
8.00 Hz * 2π = √(k/0.003 kg)
16π Hz = √(k/0.003 kg)
Square both sides to get rid of that pesky square root:
(16π Hz)² = k/0.003 kg
256π² Hz² = k/0.003 kg
Now, to find k, we simply multiply both sides by 0.003 kg:
k = 256π² Hz² * 0.003 kg
Calculating this, we find that the force constant of the spring is approximately 75.74 N/m.
So, with some bot-bemusing math, we have the answer!
To find the force constant of the spring, we can use the formula for the frequency of a mass-spring system:
f = (1/2π)√(k/m)
where f is the frequency, k is the force constant of the spring, and m is the mass attached to the spring.
Given:
Frequency (f) = 8.00 Hz
Mass (m) = 3.00 g = 0.003 kg
Substituting these values into the formula, we get:
8.00 = (1/2π)√(k/0.003)
To solve for k, we can rearrange the equation:
k = (8.00 x 2π)^2 x 0.003
Calculating this expression, we find:
k ≈ 0.753 N/m
Therefore, the force constant of the spring is approximately 0.753 N/m.
To find the force constant of the spring, we can use the formula for the frequency of vibration of an object-spring system:
f = 1 / (2π) * √(k / m)
Where:
f = frequency of vibration
k = force constant of the spring
m = mass
From the given information, we have:
f = 8.00 Hz
m = 3.00 g = 0.00300 kg
Now, rearranging the formula, we can solve for k:
k = (4π² * m * f²)
Substituting the given values:
k = (4π² * 0.00300 kg * (8.00 Hz)²)
Simplifying the equation:
k = (4 * (3.14)² * 0.00300 kg * 64.00 Hz²)
k ≈ 0.764 N/m (rounded to three decimal places)
Therefore, the force constant of the spring is approximately 0.764 N/m.
Frequency = [1/(2 pi)]*sqrt(k/m) = 8
m = 0.003 kg
Solve for k, the spring constant. It will have units of newtons per meter.
k = m*(16 pi)^2