Mercury of 10 kg, is poured into a glass U-tube with a uniform internal diameter of 1.5 cm. The system oscillates freely about its equilibrium position. Given that the density of mercury is 13600 Kg/m³ and acceleration due to gravity (g) is 9.8 m/s², Calculate the period of oscillations. Please choose the correct answer from the options below. a. 3.43 sec b. 4.04 sec C. 1.16 sec d. None of the options are true e. I do not want to answer f. 2.89 sec
Question
Mercury of 10 kg, is poured into a glass U-tube with a uniform internal diameter of 1.5 cm. The system oscillates freely about its equilibrium position. Given that the density of mercury is 13600 Kg/m³ and acceleration due to gravity (g) is 9.8 m/s², Calculate the period of oscillations. Please choose the correct answer from the options below. a. 3.43 sec b. 4.04 sec C. 1.16 sec d. None of the options are true e. I do not want to answer f. 2.89 sec
Solution
The period of oscillation for a liquid in a U-tube is given by the formula:
T = 2π √(d/gh)
where: d is the diameter of the tube, g is the acceleration due to gravity, and h is the height of the liquid column.
First, we need to find the height of the mercury column. We know the mass (m) of the mercury is 10 kg and the density (ρ) is 13600 kg/m³. We can find the volume (V) using the formula:
V = m/ρ = 10 kg / 13600 kg/m³ = 0.000735 m³
The volume of a cylinder (which the U-tube essentially is) is given by the formula:
V = πr²h
where r is the radius and h is the height. We can rearrange this to find h:
h = V / (πr²) = 0.000735 m³ / (π * (0.0075 m)²) = 4.14 m
Now we can substitute d, g, and h into the formula for T:
T = 2π √((0.015 m) / (9.8 m/s² * 4.14 m)) = 0.27 s
So, the correct answer is (d) None of the options are true.
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