on pouring 10g mercury into a test tube of mass 8g and external diameter 2cm the test tube floats vertically in water. the test tube is pressed down into the water and left. prove that the motion of the tube will be simple harmonic. also find its time period
Answers
The motion of the tube is simple harmonic and its approx time period is 0.48 sec.
Explanation:
=> It is given that,
Density of water, ρ = 1000 kg/m³
Area of the cross section of the cylinder, A = πr²
Mass of cylinder, m = [M(mercury) + M(test tube)] = [10 + 8]g = 18g = 0.018 Kg
=> Here, buoyancy force is equals to restoring force.
While pressing the test tube, suppose it is dipped by a height 'x'
thus, Restoing force = - (x*A) * ρ * g / volume
a = f/m = - A*ρ*g*x / m
But Aρg/m is constant. thus,
-α ∝ -x
Therefore, the motion of the tube is simple harmonic.
- A*ρ*g*x / m = -ω² x
ω = √π*r²*ρ*g / m
= √3.14*0.01*0.01*1000*9.8/0.018
= √ 170. 96
= 13.08 rad/sec
=> Time period , T:
T = 2π / ω
= 2 * 3.14 / 13.08
= 0.48 sec
Thus, its approx time period is 0.48 sec.
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