A spring stretches by 0.018m when a 2.8kg object is suspended from its end. How much mass should be attached to this spring so that it’s frequency of vibration is f=3.0Hz
To find the required mass, we need to use the formula for the frequency of a vibrating spring:
f = 1 / (2π) √(k / m)
where:
f is the frequency of vibration (given as 3.0 Hz),
k is the spring constant,
m is the mass attached to the spring.
We can rearrange the formula to solve for m:
m = k / (4π²f²)
We already have the frequency f and the constant k, but we need to convert the frequency from Hz to radians per second (rad/s) before plugging the values into the formula.
To convert Hz to rad/s, we multiply by 2π:
ω = 2πf
Now we can substitute the values into the formula:
m = k / (4π²ϖ²)
To calculate the spring constant k, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, this can be expressed as:
F = -kx,
where F is the force, k is the spring constant, and x is the displacement. In the given scenario, we know the displacement (0.018 m) and the mass (2.8 kg), so we can find the force using:
F = mg,
where g is the acceleration due to gravity (9.8 m/s²).
Then we can rearrange the Hooke's Law equation to solve for k:
k = -F / x.
Let's calculate the spring constant:
F = mg = 2.8 kg * 9.8 m/s² = 27.44 N.
k = -F / x = - 27.44 N / 0.018 m = -1524.4 N/m.
Now we can substitute the values into the formula for m:
m = -1524.4 N/m / (4π² * (2π * 3.0 Hz)²)
= -1524.4 N/m / (4π² * (2π * 3.0)^2)
= -1524.4 N/m / (4 * π² * (6π)²)
= -1524.4 N/m / (24π⁴)
= -1524.4 / (24π⁴) kg.
Please note that the negative sign is used because the displacement of the spring is in the opposite direction of the force applied to it, according to Hooke's Law.