A mass m at the end of a spring oscillates with a frequency of 0.83 Hz . When an additional 770 g mass is added to m, the frequency is 0.65 Hz .
Part A
What is the value of m?
Express your answer to two significant figures and include the appropriate units.
Milligrams --> Kg
To solve this problem, we need to use the relationship between the frequency of oscillation of a mass-spring system and the mass attached to the spring.
The formula for the frequency of a mass-spring system is:
f = (1 / 2π) * √(k / m)
where:
- f is the frequency of oscillation in hertz (Hz),
- k is the spring constant in newtons per meter (N/m),
- m is the mass attached to the spring in kilograms (kg), and
- π is a mathematical constant approximately equal to 3.14159.
We are given two sets of frequencies and masses:
Frequency 1: f1 = 0.83 Hz, m1 = m
Frequency 2: f2 = 0.65 Hz, m2 = m + 0.770 g
By rearranging the formula, we can solve for the unknown variable "m" in terms of the given variables and known quantities:
f = (1 / 2π) * √(k / m)
f² = (1 / 4π²) * (k / m)
4π²f² = k / m
m = k / (4π²f²)
Now we can solve for the unknown variable "m" using the given information:
1. Convert the added mass from grams to kilograms:
770 g = 770 × 10⁻³ kg = 0.770 kg
2. Substitute the values into the equation:
m₁ = k / (4π²f₁²)
m₂ = (m + 0.770 kg)
k / (4π²f₂²) = (m + 0.770 kg)
3. Equate the two equations:
k / (4π²f₁²) = (m + 0.770 kg)
4. Solve for m:
k / (4π²f₁²) - 0.770 kg = m
5. Substitute the values:
k / (4π²(0.83 Hz)²) - 0.770 kg = m
6. Calculate:
m = k / (4π²(0.83 Hz)²) - 0.770 kg
Since we don't have the value of the spring constant (k) given, we cannot directly calculate the value of "m." We need additional information or data to solve this problem.