**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 f_{s}/µ_{s}N
are correct. (We're using subscripts now. It wasn't convenient to use them in
the applet.) Here's a method.

- Set the parameters to the following: v = 10
m/s q = 30°
µ
_{s}= 0.5 g = 9.8 m/s² m = 5 kg. - Select the directions of your positive x- and y-axes. A wise choice will make the problem easier.
- Write net force equations for the x- and y-components of the forces.
- Solve the equations simultaneously for N and f
_{s}. (The teacher will be happy to show you a trick to make the algebra easy.) - Form the ratios N/mg and f
_{s}/µ_{s}N and simplify. - Substitute the values of the parameters and reduce. (We hope you didn't substitute for mass! Why?) Compare to the output values on the applet. If your values don't agree, you'll need to find your mistake before continuing.

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 v_{max}--at which
the object begins to skid. This occurs at f_{s}/µ_{s}N = 1.
That's the physics. Now you do the algebra, using your expression from A5.
Once you've solved for v_{max}, 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, v_{min}, 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 v_{min}.
Check your result on the applet.

E. Now that you've done all the algebra, this problem should be simple.
Find the ratio v_{max}/v_{min} 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?)

- What speed will the ride have to achieve in order to meet the design parameters above? (You may change the parameters if you think you have a better design.) Predict a value for the speed using the relevant equation. Then use the applet to see if the ride gives the desired N/mg (the thrill factor).

A few more questions about the design:

- How long will it take the ride to make one complete rotation at full speed?
- Before the ride starts, the riders are partially supported by the floor. At what speed is that support no longer necessary?
- What safety factor is built into the ride? That is, suppose someone is wearing very slick clothing with a friction coefficient of only 0.5. Suppose also that the ride operator is in a mischievous mood and pushes the speed up by 20%. Would riders go flying off?
- Would you ride this ride?