A mass m at the end of a spring vibrates with a frequency of 0.936 Hz. When an additional 558 g mass is added to m, the frequency is 0.634 Hz. What is the value of m?
f= 1/2pi (sqrt(k/m)
do this for both frequencies, then divide one equation by the other, solve for m.
thanks
To find the value of mass m, we can use the formula for the frequency of an object on a spring:
f = 1 / (2π) * √(k / m)
where f is the frequency, k is the spring constant, and m is the mass.
Given that frequency f₁ = 0.936 Hz and frequency f₂ = 0.634 Hz, we can set up the following equations:
0.936 = 1 / (2π) * √(k / m)
0.634 = 1 / (2π) * √(k / (m + 0.558))
Now, we can solve for m.
First, let's simplify the equations. Square both sides of both equations to eliminate the square root:
(0.936)² = (1 / (2π))^2 * (k / m)
(0.634)² = (1 / (2π))^2 * (k / (m + 0.558))
Now, divide the second equation by the first to eliminate the spring constant k:
[(0.634)² / (0.936)²] = [(k / (m + 0.558)) / (k / m)]
Simplify the equation further:
(0.634 / 0.936)² = (m / (m + 0.558))
Now, solve for m. Let's call the value on the left side of the equation A:
A = (0.634 / 0.936)²
Now, substitute A back into the equation:
A = (m / (m + 0.558))
Now we can solve for m:
A(m + 0.558) = m
Am + 0.558A = m
Am - m = -0.558A
m(A - 1) = -0.558A
m = -0.558A / (A - 1)
Now, substitute the value of A:
m = -0.558(0.634 / 0.936)² / ((0.634 / 0.936)² - 1)
Calculating the value of m gives:
m ≈ 0.311 kg
Therefore, the value of mass m is approximately 0.311 kg.