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Chapter 15: Q. 33 (page 416)
The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?
The Time Constant is 5.0
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An object in simple harmonic motion has an amplitude of , a frequency of , and a phase constant of . Draw a position graph showing two cycles of the motion.
The position of a oscillating mass is given by , where is in . Determine:
a. The amplitude.
b. The period.
c. The spring constant.
d. The phase constant.
e. The initial conditions.
f. The maximum speed.
g. The total energy.
h. The velocity at .
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?
The amplitude of an oscillator decreases toof its initial value inWhat is the value of the time constant?
Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE.
a. What is her mass if the spring constant is
b. What is her speed when the spring's length is
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