A steel sphere of radius 6.5 mm is dropped into a lake. What is the terminal speed of the sphere?
More information. Terminal speed in the water?
Yes, I think we are supposed to assume that it is in water.
The problem here is the assumptions of friction of a small sphere The drag equation is
force drag=densitywater*velocity^2*.47*area
you can calculate area from radius.
at terminal velocity, net force=dragforce
mg-bouyantforce=drag force
now the mass m of the steel ball can be calculated from density steel*volumesphere
bouyant force=densitywater*volumeball*g
do all that, and then you can calculate terminal velocity v of the ball in the water.
Thanks for your help, but I think that it is wrong. I keep getting 1.13 m/s as the answer, so I don't know if I went wrong somewhere.
To determine the terminal speed of the steel sphere, we need to consider the forces acting on the sphere when it is dropped in the lake.
The terminal speed occurs when the force of gravity pulling the sphere downward is equal to the drag force exerted by the fluid (water in this case) on the sphere, causing it to stop accelerating and reach a constant speed.
The drag force can be calculated using Stokes' law equation:
F_drag = 6πηrv
where:
F_drag is the drag force,
η (eta) is the dynamic viscosity of the fluid (water),
r is the radius of the sphere, and
v is the terminal speed.
In this case, the sphere is dropped in water, and the dynamic viscosity of water is approximately 0.001 Pa·s.
The force of gravity pulling the sphere downward can be calculated using the equation:
F_gravity = mg
where:
m is the mass of the sphere and
g is the acceleration due to gravity, approximately 9.8 m/s².
To calculate the mass of the sphere, we can use the density of steel, which is approximately 7850 kg/m³. The volume of a sphere is given by:
V = (4/3)πr³
where:
V is the volume of the sphere.
Knowing the density and the volume, we can determine the mass with the equation:
m = ρV
where:
ρ (rho) is the density.
Now, we can set the drag force equal to the force of gravity and solve for the terminal speed:
F_drag = F_gravity
6πηrv = mg
6πηr²v = ρVg
v = (ρVg) / (6πηr²)
Substituting the known values into the equation, we can calculate the terminal speed of the sphere.