Could someone please help me with my homework problem...
a) Give the equation for terminal speed in the variables from Stoke's Law, and the densities of the sphere Ps and the fluid P1.
b) Find the viscosity of motor oil (in kg/m/s) in which a steel ball of radius 0.85mm falls with a terminal speed of 4.32cm/s. The densities of the ball and oil are 7.8 and 0.78 g/ml, respectively.
https://en.wikipedia.org/wiki/Stokes%27_law
Sure! I can help you with your homework problem.
a) To find the equation for terminal speed using Stoke's Law, we need to consider the forces acting on the falling object. Stoke's Law states that the drag force on a spherical object moving through a fluid is directly proportional to its velocity and the viscosity of the fluid.
The equation for terminal speed can be derived using Stoke's Law:
Fd = 6πηrv
Where:
Fd is the drag force on the object,
η is the viscosity of the fluid,
r is the radius of the object, and
v is the velocity of the object.
Now, let's substitute the given variables:
Ps = density of the sphere = 7.8 g/ml = 7800 kg/m³ (convert grams to kilograms by dividing by 1000)
P1 = density of the fluid (motor oil) = 0.78 g/ml = 780 kg/m³ (convert grams to kilograms by dividing by 1000)
b) To find the viscosity of the motor oil, we can rearrange the equation for terminal speed as follows:
Fd = 6πηrv
Terminal speed is the maximum speed reached by the object when the drag force equals the weight of the object. Fd = Fg, where Fg is the weight of the object.
Fg = Ps * V * g
Where:
V is the volume of the sphere,
g is the acceleration due to gravity.
Now, let's plug in the given values:
r = 0.85 mm = 0.85/1000 m (convert millimeters to meters by dividing by 1000)
v = 4.32 cm/s = 4.32/100 m/s (convert centimeters to meters by dividing by 100)
Substituting these values into the equations, we can solve for η:
Fg = Fd
Ps * V * g = 6πηrv
First, let's find the volume of the sphere:
V = (4/3) * π * r³
Now, plug in the values and solve for η:
7800 * [(4/3) * π * (0.00085)^3] * 9.8 = 6πη * 0.00085 * 0.0432
Solving this equation will give us the viscosity of the motor oil (η) in kg/m/s.