Answer: Approximately 0.993 kg of mass must be added to the system to change the period of oscillation from 1.50 s to 2.05 s.
Step-by-step explanation:
To determine how much mass must be added to change the period of oscillation from 1.50 s to 2.05 s, we can use the formula for the period of a mass-spring system:
T = 2π√(m/k)
Where T is the period, m is the mass, and k is the spring constant.
Let's first calculate the initial spring constant using the given period of 1.50 s and the initial mass of 0.540 kg. Rearranging the formula, we have:
k = (4π²m) / T²
k = (4π² * 0.540 kg) / (1.50 s)²
Next, we can calculate the new mass that needs to be added to change the period to 2.05 s. Rearranging the formula again, we have:
m₂ = k(T₂²) / (4π²)
Where m₂ is the additional mass and T₂ is the desired period of 2.05 s.
m₂ = k * (2.05 s)² / (4π²)
Now, we can substitute the initial values and calculate the final mass:
m₂ = [(4π² * 0.540 kg) / (1.50 s)²] * (2.05 s)² / (4π²)
Simplifying the expression, the π² terms and s² terms cancel out, leaving:
m₂ = (0.540 kg) * (2.05 s)² / (1.50 s)²
m₂ = (0.540 kg) * (2.05)² / (1.50)²
m₂ ≈ 0.993 kg
Therefore, approximately 0.993 kg of mass must be added to the system to change the period of oscillation from 1.50 s to 2.05 s.
I hope this helps :)