A mass m at the end of a spring vibrates with a frequency of 0.87 Hz. When an additional 602 g mass is added to m, the frequency is 0.59 Hz. What is the value of m?
[1/(2 pi)]*(k/m)^1/2 = 0.87
[1/(2 pi)]*[k/(m+602)]^1/2 = 0.59
Take the ratios of the two equations
[(m+602)/m]^1/2 = 1.4746
(m+602)/m = 2.174
602/m = 1.174
m = 513 g
Well, this question is all about mass and frequency, but let me tell you a joke first to lighten the mood.
Why did the scarecrow win an award?
Because he was outstanding in his field!
Alright, let's get back to business. To solve this problem, we can use the equation that relates the mass and frequency of a vibrating spring:
f = 1 / (2π) * √(k / m)
Where f is the frequency, k is the spring constant, and m is the mass.
Now, let's set up two equations using the given information:
0.87 = 1 / (2π) * √(k / m) Equation 1
0.59 = 1 / (2π) * √(k / (m + 0.602)) Equation 2
To find the value of m, we'll need to eliminate the k term between these equations. Thankfully, the constant k is the same for both.
Now, here's another joke to keep you entertained while we solve these equations:
Why don't scientists trust atoms?
Because they make up everything!
Okay, I promise I won't interrupt again. Back to the math.
Let's simplify Equation 1 by squaring both sides:
(0.87)^2 = (1 / (2π))^2 * (k / m) Equation 1a
Now, let's simplify Equation 2 by squaring both sides as well:
(0.59)^2 = (1 / (2π))^2 * (k / (m + 0.602)) Equation 2a
Now, let's divide Equation 2a by Equation 1a:
(0.59)^2 / (0.87)^2 = (k / (m + 0.602)) / (k / m)
Simplifying further:
(0.59)^2 / (0.87)^2 = (m / (m + 0.602))
Now, we can solve for m:
(m / (m + 0.602)) = (0.59)^2 / (0.87)^2
Cross-multiplying:
(0.87)^2 * m = (0.59)^2 * (m + 0.602)
Expanding and simplifying:
0.7599 * m = 0.3481 * (m + 0.602)
Distributing:
0.7599 * m = 0.3481 * m + 0.2094
Simplifying further:
0.4118 * m = 0.2094
Now, let's solve for m:
m = 0.2094 / 0.4118
m ≈ 0.508 kg
So, the value of m is approximately 0.508 kg.
I hope my answer added some laughter to your day! If you have any more questions or need further assistance, feel free to ask!
To find the value of mass m, we can start by using the formula for the frequency of a mass-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 denote the original mass m as m1 and the added mass as m2.
Given that the frequency with just m1 is 0.87 Hz and the frequency with (m1 + m2) is 0.59 Hz, we can set up the equations as follows:
0.87 = 1 / (2π) * sqrt(k / m1)
0.59 = 1 / (2π) * sqrt(k / (m1 + m2))
Now, we can solve these two equations simultaneously to find the values of m1 and m2. Let's start by squaring both equations:
(0.87)^2 = (1 / (2π))^2 * (k / m1)
(0.59)^2 = (1 / (2π))^2 * (k / (m1 + m2))
Now, we can eliminate the spring constant k by dividing the two equations:
(0.87)^2 / (0.59)^2 = (k / m1) / (k / (m1 + m2))
(0.87)^2 / (0.59)^2 = m1 / (m1 + m2)
Now, we can simplify the equation:
(0.87)^2 / (0.59)^2 = m1 / (m1 + m2)
0.7744 / 0.3481 = m1 / (m1 + m2)
2.2240 ≈ m1 / (m1 + m2)
Cross-multiplying:
2.2240 * (m1 + m2) ≈ m1
2.2240 * m1 + 2.2240 * m2 ≈ m1
2.2240 * m2 ≈ -1.2240 * m1
Dividing by m1:
2.2240 * m2 / m1 ≈ -1.2240
m2 / m1 ≈ -1.2240 / 2.2240
m2 / m1 ≈ -0.5509
Now, let's solve for m1:
m2 / m1 ≈ -0.5509
m1 * m2 / m1 ≈ -0.5509 * m1
m2 ≈ -0.5509 * m1
m1 ≈ m2 / -0.5509
Substituting the given mass m2 = 602 g:
m1 ≈ 602 g / -0.5509
m1 ≈ -1092.51 g
Since mass cannot be negative, we discard the negative value and take the positive value:
m1 ≈ 1092.51 g
Therefore, the value of mass m, which is m1, is approximately 1092.51 grams.
To solve this problem, we need to use the concept of simple harmonic motion and the equation for the frequency of a mass-spring system. The frequency (f) of a mass-spring system is related to the mass of the object (m) and the spring constant (k) by the equation:
f = (1 / 2π) * √(k / m)
We are given two sets of data:
1. When only mass m is attached to the spring, the frequency is 0.87 Hz.
2. When an additional mass of 602 g is added to mass m, the frequency becomes 0.59 Hz.
Let's denote the initial mass m as M (in kilograms), and the final mass (M + 0.602 kg) as M'.
Using these values, we can set up two equations:
For the first case: f = 0.87 Hz
0.87 Hz = (1 / 2π) * √(k / M)
For the second case: f = 0.59 Hz
0.59 Hz = (1 / 2π) * √(k / (M + 0.602))
Divide these two equations:
(0.87 Hz) / (0.59 Hz) = (√(k / M)) / (√(k / (M + 0.602)))
Simplify:
0.87 / 0.59 = √((M + 0.602) / M)
Square both sides of the equation:
(0.87 / 0.59)² = (M + 0.602) / M
Solve for M:
M = (0.87 / 0.59)² * (M + 0.602)
M = (0.87 / 0.59)² * M + (0.87 / 0.59)² * 0.602
M - (0.87 / 0.59)² * M = (0.87 / 0.59)² * 0.602
M * (1 - (0.87 / 0.59)²) = (0.87 / 0.59)² * 0.602
M = ((0.87 / 0.59)² * 0.602) / (1 - (0.87 / 0.59)²)
Now we can substitute the values and calculate:
M = ((0.87 / 0.59)² * 0.602) / (1 - (0.87 / 0.59)²)
M ≈ 0.081 kg
Therefore, the value of m is approximately 0.081 kg.