A hydrometer is a device that measures the density of a liquid. The one shown in the figure has a cylindrical bulb of radius R attached to a cylindrical stem of radius r and length l. When placed in a liquid, the device floats as shown in the figure with a length h of stem protruding. Given that the mass of the hydrometer is M, find the density ρ of the liquid. Express your answer in terms of M ,R, r ,l and h (enter M for M, R for R, r for r, l for l, h for h and pi for π).
ρ=
M/(pi*r^2*(l-h) + 4*pi*R^3/3)
thanks
To find the density of the liquid (ρ) using the given information about the hydrometer, we can make use of the principle of buoyancy.
The buoyant force acting on the hydrometer is equal to the weight of the volume of liquid displaced by the submerged part of the hydrometer.
First, let's find the volume of the submerged part of the hydrometer.
The submerged part consists of a cylinder with radius r and length h. The volume V of the cylinder can be calculated as follows:
V = πr^2h
Next, let's find the weight of the volume of liquid displaced.
The weight W can be calculated using the following equation:
W = Mg
Where M is the mass of the hydrometer and g is the acceleration due to gravity.
Finally, we can find the density of the liquid ρ by dividing the weight W by the volume V:
ρ = W / V = Mg / (πr^2h)
Therefore, the density of the liquid ρ is given by Mg / (πr^2h).