8. Centripetal Forces & Gravitation

A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is spherical.

A small car with mass $0.800$ kg travels at constant speed on the inside of a track that is a vertical circle with radius $5.00$ m (Fig. E$5.45$). If the normal force exerted by the track on the car when it is at the top of the track (point $B$) is $6.00$ N, what is the normal force on the car when it is at the bottom of the track (point $A$)?

For safety, elevators have a rotational governor, a device that is attached to and rotates with one of the elevator's pulleys. The governor, shown in FIGURE P8.63, is a disk with two hollow channels holding springs with metal blocks of mass m attached to their free ends. The faster the governor spins, the more the springs stretch. At a critical angular velocity ω_c, the metal blocks contact the housing, which completes a circuit and activates an emergency brake. The spring force on a mass, which we will explore more thoroughly in Chapter 9, is F_Sp = k(r - L), where k is the spring constant measured in N/m, and L is the relaxed (unstretched) length of the spring. Suppose a rotational governor has L = 0.80R and the emergency brake activates when the metal blocks reach r = R. What is the critical angular velocity in rpm if R = 15cm, k = 20 N/m, and m = 25g? Ignore gravity.

The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of $100$ m. Its name comes from its $60$ arms, each of which can function as a second hand (so that it makes one revolution every $60.0$ s). A passenger weighs $882$ N at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel?

(II) If a plant is allowed to grow from seed on a rotating platform, it will grow at an angle, pointing inward. Calculate what this angle will be (put yourself in the rotating frame) in terms of g, r, and ω. Why does it grow inward rather than outward?

If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.72, 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 z = (ω² / 2g) r². 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.

An object of mass m swings in a horizontal circle on a string of length L that tilts downward at angle θ. Find an expression for the angular velocity ω.