Chapter 8: Q.21 (page 199)
The normal force equals the magnitude of the gravitational force as a roller coaster car crosses the top of a diameter loop-the-loop. What is the car’s speed at the top?
The speed of the roller coaster at the top is
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If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.71, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equation
Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.
A motorcycle daredevil plans to ride up a -high, ramp, sail across a -wide pool filled with hungry crocodiles, and land at ground level on the other side. He has done this stunt many times and approaches it with confidence. Unfortunately, the motorcycle engine dies just as he starts up the ramp. He is going at that instant, and the rolling friction of his rubber tires (coefficient ) is not negligible. Does he survive, or does he become crocodile food? Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp.
The father of Example 8.2 stands at the summit of a conical hill as he spins his 20 kg child around on a 5.0 kg cart with a 2.0-m-long rope. The sides of the hill are inclined at 20o. He
again keeps the rope parallel to the ground, and friction is negligible. What rope tension will allow the cart to spin with the same 14 rpm it had in the example?
You’ve taken your neighbor’s young child to the carnival to ride the rides. She wants to ride The Rocket. Eight rocket-shaped cars hang by chains from the outside edge of a large steel disk. A vertical axle through the center of the ride turns the disk, causing the cars to revolve in a circle. You’ve just finished taking physics, so you decide to figure out the speed of the cars while you wait. You estimate that the disk is in diameter and the chains are long. The ride takes to reach full speed, then the cars swing out until the chains are from vertical. What is the cars’ speed?
The physics of circular motion sets an upper limit to the speed of human walking. (If you need to go faster, your gait changes from a walk to a run.) If you take a few steps and watch
what’s happening, you’ll see that your body pivots in circular motion over your forward foot as you bring your rear foot forward for the next step. As you do so, the normal force of the
ground on your foot decreases and your body tries to “lift off” from the ground.
a. A person’s center of mass is very near the hips, at the top of the legs. Model a person as a particle of mass m at the top of a leg of length L. Find an expression for the person’s maximum walking speed vmax.
b. Evaluate your expression for the maximum walking speed of a 70 kg person with a typical leg length of 70 cm. Give your answer in both m/s and mph, then comment, based on your
experience, as to whether this is a reasonable result. A “normal” walking speed is about 3 mph.
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