A hydrometer is a device that measures the density of a liquid. The one shown in the figure has a cylindrical bulb of radius attached to a cylindrical stem of radius and length . When placed in a liquid, the device floats as shown in the figure with a length of stem protruding. Given that the mass of the hydrometer is , find the density of the liquid. Express your answer in terms of , , , and (enter M for , R for , r for , l for , h for and pi for ).
display.cgi?id=1386766366
To find the density of the liquid, we need to consider the forces acting on the hydrometer when it is floating in the liquid.
First, let's consider the buoyant force. The buoyant force is equal to the weight of the liquid displaced by the submerged part of the hydrometer. Since the hydrometer is floating, the buoyant force is equal to the weight of the hydrometer.
The weight of the hydrometer can be calculated as the mass of the hydrometer multiplied by the acceleration due to gravity. So, the weight of the hydrometer is:
Weight = Mass * gravitational acceleration
Weight = M * g
Next, let's consider the weight of the liquid displaced by the submerged part of the hydrometer. The liquid displaced has a volume equal to the volume of the submerged part of the hydrometer. The volume of the cylinder is given by the formula:
Volume = pi * r^2 * l
Where r is the radius of the cylindrical stem and l is the length of the submerged part of the hydrometer.
The weight of the liquid displaced is given by:
Weight of displaced liquid = Density of liquid * Volume of displaced liquid * gravitational acceleration
Since the volume of the displaced liquid is the same as the volume of the submerged part of the hydrometer, we can rewrite the equation as:
Density of liquid * Volume of submerged part * gravitational acceleration = Mass of hydrometer * gravitational acceleration
Now, let's rearrange the equation to solve for the density of the liquid:
Density of liquid = (Mass of hydrometer * gravitational acceleration) / (Volume of submerged part * gravitational acceleration)
The gravitational acceleration and the gravitational acceleration in the numerator and denominator cancel out, so the equation becomes:
Density of liquid = Mass of hydrometer / Volume of submerged part
Substituting the values given in the question with the respective variables, we get:
Density of liquid = M / (pi * r^2 * l)
So, the density of the liquid is M / (pi * r^2 * l).