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.

a) The equation for terminal speed in Stoke's Law is given by:

v = (2/9) * ((d^2) * g * (Ps - P1) / η)

Where:
v is the terminal speed,
d is the diameter of the object (in this case, the diameter of the sphere),
g is the acceleration due to gravity,
Ps is the density of the sphere, and
P1 is the density of the fluid.

b) To find the viscosity (η) of motor oil, we can rearrange the equation as follows:

η = (2/9) * ((d^2) * g * (Ps - P1) / v)

Given data:
d (diameter of the sphere) = 0.85 mm = 0.85 × 10^(-3) m (convert from mm to m)
g (acceleration due to gravity) = 9.8 m/s^2
Ps (density of the sphere) = 7.8 g/ml = 7.8 × 10^3 kg/m^3 (convert from g/ml to kg/m^3)
P1 (density of the fluid) = 0.78 g/ml = 0.78 × 10^3 kg/m^3 (convert from g/ml to kg/m^3)
v (terminal speed) = 4.32 cm/s = 4.32 × 10^(-2) m/s (convert from cm/s to m/s)

Now substitute the given values into the equation and solve for η:

η = (2/9) * ((0.85 × 10^(-3))^2 * 9.8 * (7.8 × 10^3 - 0.78 × 10^3) / (4.32 × 10^(-2)))

η ≈ 0.067 kg/m/s

Therefore, the viscosity of the motor oil is approximately 0.067 kg/m/s.