A mass m at the end of a spring vibrates with a frequency of 0.92 Hz. When an additional 600 g mass is added to m, the frequency is 0.64 Hz. What is the value of m?
Start with the relation
f = [1/(2 pi)]sqrt(m/k)
The spring constant k remains constant
Since the frequency decreases by a factor of 1.4375, the mass must have increased by a factor 1.4375^2, which is 2.0664
Let M be the original mass that gave the 0.92 Hz frequency.
(M + 600)/600 = 2.0664
M/600 = 1.064
M = 638 g
You could also have solved this with two simultaneous equations and algebra, but this way is easier
that's not right, I think I'm gonna have to use 2 equations but i'm not sure how to do it.
I was careless and should have written
f = [1/(2 pi)] sqrt(k/m), but the following logic and final number I believe to be correct.
hmm Its not working out. this isnt the actual problem but one of the practice ones (online homework) but i keep trying this method with different numbers and they aren't working out. Do you know how to explain the way to do it with 2 equations?
0.92 = [1/(2 pi)] sqrt(k/m)
0.64 = [1/(2 pi)] sqrt[k/(m+0.6)],
where 0.6 is the added mass in kg.
Now divide the first equation by the second, to get rid of the unknown k
1.4375 = sqrt [(m + 0.6)/m]
2.0664 = (m + 0.6)/m = 1 + 0.6/m
1.0664 = 0.6/m
m = 0.563 kg
I made an algebraic mistake intitally, and should have written
(M + 600)/M = 2.0664
I was working with M in grams, which is OK when dealing with ratios, but I had the wrong denominator in the last equation.
So I not only wrote the original k/m ratio upside down, but also later made an algebraic error. I regret both errors and the confusion it caused you. I should have been more careful
To find the value of m, we need to use the formula for the frequency of a mass-spring system:
f = (1/2π) * √(k/m)
Where f is the frequency, k is the spring constant, and m is the mass.
First, let's assume the initial mass (m) is what we are trying to find. We'll call it m1.
Using the given initial frequency of 0.92 Hz:
0.92 = (1/2π) * √(k/m1)
Next, a 600 g mass (0.6 kg) is added to m1, creating a new total mass (m2), which we need to find.
Using the new frequency of 0.64 Hz:
0.64 = (1/2π) * √(k/m2)
Now, we have two equations and two unknowns (m1 and m2). Let's solve these equations simultaneously to find the values of m1 and m2.
From the first equation, we can isolate √(k/m1):
√(k/m1) = (0.92 * 2π)
Squaring both sides, we get:
k/m1 = (0.92 * 2π)^2
Similarly, from the second equation:
k/m2 = (0.64 * 2π)^2
Now, we need to eliminate k from the equations. Since the spring constant k remains the same in both cases, we can set the two expressions for k/m equal to each other:
k/m1 = k/m2
(0.92 * 2π)^2 = (0.64 * 2π)^2
Simplifying:
(0.92)^2 = (0.64)^2
0.8464 = 0.4096
Since this is not true, we made an incorrect assumption that the spring constant remained the same. However, the spring constant does not change when additional mass is added.
So, let's find the value of the spring constant k using the first equation:
k/m1 = (0.92 * 2π)^2
Simplifying further:
k/m1 = 3.27008
Now, let's use this value of k in the second equation:
k/m2 = (0.64 * 2π)^2
Since k/m1 = k/m2, we can set these equations equal to each other:
k/m1 = k/m2
3.27008 = (0.64 * 2π)^2
Simplifying:
3.27008 = 0.4096 * 4π^2
Divide both sides by 0.4096 * 4π^2:
m2 = 3.27008 / (0.4096 * 4π^2)
Calculating:
m2 ≈ 10.0 kg
Therefore, the value of m is approximately 10.0 kg.