Question

You have a glass ball with a radius of 2.00 mm and a density of 2500 kg/m3. You hold the ball so it is fully submerged, just below the surface, in a tall cylinder full of glycerin, and then release the ball from rest. Take the viscosity of glycerin to be 1.5 Pa s and the density of glycerin to be 1250 kg/m3. Use g = 10 N/kg = 10 m/s2. Also, note that the drag force on a ball moving through a fluid is:

Fdrag = 6πηrv .

(a) Note that initially the ball is at rest. Sketch (to scale) the free-body diagram of the ball just after it is released, while its velocity is negligible.

(b) Calculate the magnitude of the ball’s initial acceleration.

(c) Eventually, the ball reaches a terminal (constant) velocity. Sketch (to scale) the free-body diagram of the ball when it is moving at its terminal velocity.

(d) Calculate the magnitude of the terminal velocity.

(e) What is the magnitude of the ball’s acceleration, when the ball reaches terminal velocity?

(f) Let’s say that the force of gravity acting on the ball is 4F, directed down. We can then express all the forces in terms of F. (For instance, you might label a force on a free-body diagram as “Fdrag = 3F”.) Sketch three free-body diagrams, and express all forces in terms of F. Hint: do the middle one last.

Initial FBD FBD for when v = half FBD for when (when released from rest) the terminal velocity v = terminal velocity

(g) How does the net force in the left-most free-body diagram compare to that in the middle free-body diagram? Combine that information with your result from problem 1, regarding the initial acceleration, to find the magnitude of the acceleration when the ball’s speed is half the terminal speed.

(h) Take down to be positive for all your graphs. (i) Sketch a grapch of the ball’s acceleration as a function of time; (ii) sketch a graph of the ball’s velocity as a function of time; (iii) sketch a graph of the ball’s position as a function of time. If you can be quantitative with your axis labels, then do so.

Answer 1