Q.6 - Excercises And ProblemsExpert-verified
What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in
c. phase constant
From the given amplitude, frequency and phase constant.
The amplitude is the length of the peak from the time axis. From , it is clear that the amplitude is
The frequency is simply the number of waves per unit time. From , in the number of waves completed is
Therefore, frequency is,
Suppose the equation of the wave is,
Here, is the angular frequency of the wave, and is the phase constant.
The time period of the wave is,
Therefore, the angular frequency is,
Let's take what happens at
The equation takes the form,
Therefore, the initial phase constant or the initial phase is
Suppose a large spherical object, such as a planet, with radius and mass has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius ; there is no net gravitational force from the mass in the spherical shell with .
. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of , , , , and any necessary constants.
. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with . Suppose an intrepid astronaut exploring a -km-diameter, kg asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?
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