Problems for Physics with Calculus Students
A. This first problem is one of verification. That is, given the values of the various input parameters, show that the output values for N/mg and fs/µsN are correct. (We're using subscripts now. It wasn't convenient to use them in the applet.) Here's a method.
B. This is a problem requiring prediction. Predict the design speed for the curve (same q and g as above). Do the relevant net force problem, solve for v, and substitute parameters to get a value. Enter the speed in the applet and see if the friction force disappears.
C. This problem is strictly algebra. Well, almost. It starts with a bit of physics. You're to predict the speed--we'll call it vmax--at which the object begins to skid. This occurs at fs/µsN = 1. That's the physics. Now you do the algebra, using your expression from A5. Once you've solved for vmax, substitute parameters to obtain a value. Then check it on the applet.
D. You learned from the preliminary explorations that there's also a minimum speed, vmin, below which the object begins sliding toward the center of the curve. The net force problem for this is similar to the one you did in part A. However, there will be some sign changes, since the friction force points in the opposite direction. Repeat part A2-5 for this case. Then find vmin. Check your result on the applet.
E. Now that you've done all the algebra, this problem should be simple. Find the ratio vmax/vmin for a banking angle of 45°. Keep the same friction coefficient as above.
F. Now put yourself in the position of an amusement park ride designer. Your job is to design a circular ride with slanted walls against which the riders lean. It's like being on a banked curve, except that the roadbed moves with the object. The physics is the same. You want the riders to have a thrill and this means achieving an apparent weight of about 3 mg's. You want the walls to be nearly vertical, say 60°. Otherwise, the riders could experience some serious pooling of blood in their heads. The riders won't be strapped in, since that would reduce the thrill. That means the friction force of their clothing against the wall has to be enough to keep them from sliding up and over the wall. That could be bad for business. So you'd better have a friction coefficient of at least 0.7. Now 10 m might be a bit large for the radius of the ride; that's a diameter of almost 70 feet. However, we're ambitious and figure we can make money faster by accommodating more passengers at once. (Besides, the applet is stuck at 10 m and you want to use it to check your design, don't you?)
A few more questions about the design: