Practice Exam (Chap. 12, 13, 15)This is a practice exam for Ch. 12, 13, and 15, although the final exam will cover Ch. 1 throughCh. 11.1. A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite.If a geosynchronous satellite is in an equatorial orbit, it position appears stationary with respectto a ground station, and it is known as a geostationary satellite. Find the radius R (the distancefrom the center of the earth) of the orbit of a geostationary satellite that circles the earth. Usethat G = 6.67 × 10−11N m2/kg2, and the earth has the mass mE= 5.98 × 1024kg and radiusRE= 6.38 × 106m.2. (a) Two identical satellites orbit the earth in stable orbits. One satellite orbits with a speedv at a distance r from the center of the earth. The second satellite travels at a speed v2that isless than v at a distance r2from the center of the earth. Is r2larger than r?(b) Now assume that a satellite of mass m is orbiting the earth at a distance r from the centerof the earth with speed ve. An identical satellite is orbiting the moon at the same distance witha speed vm. How does the time Tmit takes the satellite circling the moon to make one revolutioncompare to the time Teit takes the satellite orbiting the earth to make one revolution? That is, isTmlarger than Te?3. Your boss at the Cut-Rate Cuckoo Clock Company wants to save money on materials. Youare asked to reduce the balance wheel, which is cylindrical, dimensions to one-third their originalvalues. Assume the new wheel has the same density and the same coil spring (thus the same torsionconstant) as the original.(a) By what factor would the frequency of the angular simple harmonic motion of the balancewheel change?(b) By what factor would the torsion constant need to be changed to make the smaller balancewheel oscillate at the original frequency?4. A thin metal disk of mass m = 2.00 × 10−3kg and radius R = 2.20 cm is attached at itscenter to a long fiber. When the disk is turned from the relaxed state through a small angle θ, thetorque exerted by the fiber on the disk is proportional to θτ = −kθ.The constant k is called the torsional constant and is a property of the fiber.(a) Find an expression of k in terms of moment of inertia I of the disk and the angular frequencyω of small, free oscillations.(b) The disk, when twisted and released, oscillates with a period T of 1.00 s. Find the torsionalconstant k of the fiber. The moment of inertia of the disk is12mR2.5. An oscillator creates periodic waves on a stretched string.(a) If the period of the oscillator doubles, what happens to the wavelength and wave speed?(b) If the amplitude of the oscillator doubles, what happens to the wavelength and wave speed?6. Wave motion is characterized by two velocities: the velocity with which the wave moves inthe medium (e.g., air or a string) and the velocity of the medium (the air or the string itself).Consider a transverse wave traveling in a string. The mathematical form of the wave isy(x, t) = A sin(kx − ωt).(a) Find the velocity of propagation vpof this wave.(b) Find the y velocity vyof a point on the string as a function of x and t.(c) What is the x component of the velocity vxof the string?(d) Find the slope of the string ∂y(x, t)/∂x as a function of position x and
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