You have a glass ball with a radius of 2.00 mm and a density of2500 kg/m3. You hold the ball so it is fully submerged, just belowthe surface, in a tall cylinder full of glycerin, and then releasethe ball from rest. Take the viscosity of glycerin to be 1.5 Pa sand the density of glycerin to be 1250 kg/m3. Use g = 10 N/kg = 10m/s2. Also, note that the drag force on a ball moving through afluid 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, whileits velocity is negligible.
(b) Calculate the magnitude of the ball’s initialacceleration.
(c) Eventually, the ball reaches a terminal (constant) velocity.Sketch (to scale) the free-body diagram of the ball when it ismoving at its terminal velocity.
(d) Calculate the magnitude of the terminal velocity.
(e) What is the magnitude of the ball’s acceleration, when theball reaches terminal velocity?
(f) Let’s say that the force of gravity acting on the ball is4F, directed down. We can then express all the forces in terms ofF. (For instance, you might label a force on a free-body diagram as“Fdrag = 3Fâ€.) Sketch three free-body diagrams, and express allforces in terms of F. Hint: do the middle one last.
Initial FBD FBD for when v = half FBD for when (when releasedfrom rest) the terminal velocity v = terminal velocity
(g) How does the net force in the left-most free-body diagramcompare to that in the middle free-body diagram? Combine thatinformation with your result from problem 1, regarding the initialacceleration, to find the magnitude of the acceleration when theball’s speed is half the terminal speed.
(h) Take down to be positive for all your graphs. (i) Sketch agrapch 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 youcan be quantitative with your axis labels, then do so.
