Answer :
The unknown mass is approximately 0.903 kg, and the spring constant is about 125663.7056 N/m.
To solve this problem, we can use the formula for the natural frequency of a mass-spring system:
[tex]\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \][/tex]
Where:
- [tex]\( f \)[/tex] = natural frequency (in cycles per minute)
- [tex]\( k \)[/tex] = spring constant (in Newtons per meter, N/m)
- [tex]\( m \)[/tex] = mass (in kilograms, kg)
Given that the natural frequency [tex]\( f \)[/tex] changes when a mass [tex]\( m \)[/tex] is added, we can set up two equations using the information provided:
1. When the unknown mass is attached:
[tex]\[ 94 = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \][/tex]
2. When the unknown mass plus 0.453 kg is attached:
[tex]\[ 76.7 = \frac{1}{2\pi} \sqrt{\frac{k}{(m + 0.453)}} \][/tex]
We have two equations and two unknowns [tex](\( m \)[/tex] and [tex]\( k \))[/tex]. Let's solve this system of equations.
First, let's rewrite the equations to solve for [tex]\( k \)[/tex] in terms of [tex]\( m \)[/tex]:
1. [tex]\( k = (2\pi)^2 \times 94^2 \times m \)[/tex]
2. [tex]\( k = (2\pi)^2 \times 76.7^2 \times (m + 0.453) \)[/tex]
Now, equate the expressions for [tex]\( k \)[/tex]:
[tex]\[ (2\pi)^2 \times 94^2 \times m = (2\pi)^2 \times 76.7^2 \times (m + 0.453) \][/tex]
Now, we can solve for [tex]\( m \)[/tex]:
[tex]\[ 94^2 \times m = 76.7^2 \times (m + 0.453) \][/tex]
[tex]\[ 94^2 \times m = 76.7^2 \times m + 76.7^2 \times 0.453 \][/tex]
[tex]\[ 94^2 \times m - 76.7^2 \times m = 76.7^2 \times 0.453 \][/tex]
[tex]\[ m (94^2 - 76.7^2) = 76.7^2 \times 0.453 \][/tex]
[tex]\[ m = \frac{76.7^2 \times 0.453}{94^2 - 76.7^2} \][/tex]
Now, let's calculate [tex]\( m \)[/tex]:
[tex]\[ m \approx \frac{(76.7)^2 \times 0.453}{(94)^2 - (76.7)^2} \][/tex]
[tex]\[ m \approx \frac{(76.7)^2 \times 0.453}{8836 - 5885.69} \][/tex]
[tex]\[ m \approx \frac{(76.7)^2 \times 0.453}{2950.31} \][/tex]
[tex]\[ m \approx \frac{(5885.69) \times 0.453}{2950.31} \][/tex]
[tex]\[ m \approx \frac{2664.33}{2950.31} \][/tex]
[tex]\[ m \approx 0.903 \, \text{kg} \][/tex]
Now that we have found the value of [tex]\( m \)[/tex], we can substitute it into one of the equations to find [tex]\( k \)[/tex]. Let's use the first equation:
[tex]\[ k = (2\pi)^2 \times 94^2 \times m \][/tex]
[tex]\[ k = (2\pi)^2 \times (94)^2 \times 0.903 \][/tex]
[tex]\[ k \approx 4\pi^2 \times 94^2 \times 0.903 \][/tex]
[tex]\[ k \approx 4\pi^2 \times 8836 \times 0.903 \][/tex]
[tex]\[ k \approx 4\pi^2 \times 7977.708 \][/tex]
[tex]\[ k \approx 4 \times 9.8696 \times 7977.708 \][/tex]
[tex]\[ k \approx 125663.7056 \, \text{N/m} \][/tex]
