The string is 0.95 m long with tension 8.00 N. The total mass of the string is 12.34 g. Find the period of the oscillation.
To find the period of the oscillation, we must first determine the effective length of the string. The formula for the period of an oscillating string is given by:
T = 2π * √(L / g)
Where:
T = Period of oscillation
L = Effective length of the string
g = Acceleration due to gravity
Now let's calculate the effective length of the string. The effective length accounts for the tension in the string, which causes it to stretch and change its original length. The effective length can be calculated using the formula:
L = L0 + ΔL
Where:
L0 = Original length of the string
ΔL = Change in length of the string due to tension
In this case, the original length of the string (L0) is given as 0.95 m. To determine ΔL, we can use Hooke's Law, which states that the change in length of a spring or string is directly proportional to the applied force (tension) and inversely proportional to its stiffness (spring constant):
ΔL = (Tension * L0) / (Stiffness)
In this case, the tension (T) is given as 8.00 N, and the total mass of the string is given as 12.34 g. To find the stiffness (k), we need to convert the mass to kilograms and use the formula:
Stiffness (k) = (mass * g) / L0
Where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Now, let's calculate the effective length (L):
L = L0 + ΔL
L = L0 + (Tension * L0) / (Stiffness)
Next, we can substitute the given values into the formula to calculate L and then substitute it into the formula for the period T:
L = 0.95 m + ((8.00 N * 0.95 m) / ((12.34 g * 9.8 m/s^2) / 0.001 kg/g))
L ≈ 0.95 m + 0.611 m ≈ 1.561 m
Finally, we can substitute the value of L into the formula for the period T:
T = 2π * √(L / g)
T = 2π * √(1.561 m / 9.8 m/s^2)
T ≈ 2π * √(0.159 m)
T ≈ 2π * 0.399 m ≈ 2.51 s
Therefore, the period of oscillation of the string is approximately 2.51 seconds.