A spring vibrates at 8Hz,when a mass of 0.82kg is hung from it.What is the spring constant?
Isn't this a standard equation?
period= 1/frequency= 2pi Sqrt (mass/k) check that.
To find the spring constant of a vibrating spring, we can use Hooke's Law and the equation for the frequency of a vibrating spring.
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement x from its equilibrium position, and the constant of proportionality is the spring constant k. Mathematically, this can be expressed as:
F = -kx
The equation for the frequency of a vibrating spring with mass m attached to it is given by:
f = (1/2π) √(k/m)
Given that the spring vibrates at a frequency of 8 Hz and a mass of 0.82 kg is hung from it, we can rearrange the equation to solve for the spring constant:
8 = (1/2π) √(k/0.82)
First, let's rearrange the equation to solve for k:
8 * 2π = √(k/0.82)
Square both sides of the equation to eliminate the square root:
(8 * 2π)^2 = k/0.82
k = (8 * 2π)^2 * 0.82
Now, we can calculate k:
k ≈ 976.12 N/m
Therefore, the spring constant is approximately 976.12 N/m.
To determine the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is:
F = -k * x
Where:
- F is the force exerted by the spring (in newtons).
- k is the spring constant (in newtons per meter).
- x is the displacement of the spring from its equilibrium position (in meters).
In this case, we are given that the spring vibrates at a frequency of 8 Hz and a mass of 0.82 kg is hung from it.
The frequency of vibration (f) of a mass-spring system can be calculated using the formula:
f = (1 / (2π)) * sqrt(k / m)
Where:
- f is the frequency of vibration (in hertz).
- k is the spring constant (in newtons per meter).
- m is the mass of the object (in kilograms).
- π is a mathematical constant, approximately equal to 3.14159.
Rearranging the formula, we can solve for k:
k = (4π^2 * m * f^2)
Now, we can substitute the known values into the equation:
k = (4π^2 * 0.82 * 8^2)
Calculating the equation:
k ≈ 162.57 N/m
Therefore, the spring constant is approximately 162.57 N/m.