When an object of mass m1 is hung on a vertical spring and set into vertical simple harmonic motion, its frequency is 12Hz. When another object of mass m2 is hung on the spring along with m1, the frequency of the motion is 4Hz. Find the ratio m2/m1 of the masses.
To find the ratio m2/m1 of the masses, we can use the formula for the frequency of a vertically oscillating spring system:
f = 1 / (2π) * sqrt(k / m)
where f is the frequency, k is the spring constant, and m is the mass.
Let's assume the mass of the first object is m1, and the mass of the second object is m2. When only m1 is hung on the spring, its frequency f1 is 12Hz. When both m1 and m2 are hung on the spring together, the frequency f2 is 4Hz.
We can set up the following equations based on the information given:
f1 = 1 / (2π) * sqrt(k / m1)
f2 = 1 / (2π) * sqrt(k / (m1 + m2))
To find the ratio m2/m1, we can rearrange the equations:
sqrt(k / m1) = 2π * f1
sqrt(k / (m1 + m2)) = 2π * f2
We can now square both equations to eliminate the square root:
k / m1 = (2π * f1)^2
k / (m1 + m2) = (2π * f2)^2
Now we can solve for (m2 + m1) / m1 by dividing the second equation by the first equation:
(m2 + m1) / m1 = (k / (m1 + m2)) / (k / m1)
(m2 + m1) / m1 = (2π * f2)^2 / (2π * f1)^2
(m2 + m1) / m1 = (2π * f2)^2 * (1 / (2π * f1)^2)
(m2 + m1) / m1 = (f2 / f1)^2
(m2 + m1) / m1 = (4 / 12)^2
(m2 + m1) / m1 = (1/3)^2
(m2 + m1) / m1 = 1/9
Simplifying the equation, we get:
m2 + m1 = m1/9
m2 = m1/9 - m1
m2 = m1 * (1/9 - 1)
m2 = m1 * (8/9)
Therefore, the ratio m2/m1 of the masses is 8/9.