Small spheres of diameter 1.00mm fall through 20 degree Celsius water at a terminal speed of 1.10 cm/s. calculate the density of the spheres.

1020 kg/m3

To calculate the density of the spheres, we can use the concept of terminal velocity. The terminal velocity is the maximum constant speed reached by an object when the force of gravity is balanced by an opposing force, such as drag or buoyancy.

In this case, the opposing force is the drag force experienced by the spheres as they fall through the water. For small spheres moving through a fluid, the drag force can be given by Stokes' Law:

F_drag = 6πηrv

where F_drag is the drag force, η is the dynamic viscosity of the fluid, r is the radius of the spheres, and v is the velocity of the spheres.

We can rearrange this equation to solve for the radius of the spheres:

r = F_drag / (6πηv)

Given the diameter of the spheres (1.00 mm), we can convert it to radius:

r = (1.00 mm) / 2 = 0.50 mm = 0.050 cm

Now, we can substitute the known values into the equation:

0.050 cm = F_drag / (6πη * 1.10 cm/s)

Solving for the drag force:

F_drag = 0.050 cm * 6πη * 1.10 cm/s

Next, we need to find the mass of the spheres. The mass can be calculated using the formula:

mass = density * volume

The volume of a sphere can be calculated using the formula:

volume = (4/3) * π * r^3

Substituting the known values:

volume = (4/3) * π * (0.050 cm)^3

Now we can calculate the mass:

mass = density * (4/3) * π * (0.050 cm)^3

Finally, we can equate the gravitational force (mg) with the drag force (F_drag):

mg = F_drag

where g is the acceleration due to gravity (9.81 m/s^2).

Substituting the known values:

(density * volume) * g = 0.050 cm * 6πη * 1.10 cm/s

Now, we can solve for the density:

density = (0.050 cm * 6πη * 1.10 cm/s) / [(4/3) * π * (0.050 cm)^3 * g]

Plugging in the values, such as the dynamic viscosity of water at 20 degrees Celsius, and converting units, will give you the density of the spheres.