The gravitational force on a body located a distance R from the center of a uniform spherical mass is due solely to the mass lying at distance r<R or r=R, measured from the center of the sphere. This mass exerts a force as if it were a point mass at the origin.
Use the above result to show that if you drill a hole through the earth and then fall in, you will execute simple harmonicmotion about the earth's center. Find the time it takes you to return to your point of departure and show that this is the time needed for a satellite to circle the earth in low orbit with r approximately equaling R(earth). In deriving this result, you need to treat the earth as a uniformly dense sphere, and you must neglect all friction and any effects due to the earth's rotation.

Solution Preview

First we have to find the formula for the force due to gravity on a mass m at distance r from the center of the Earth of mass M and radius R. Since the Earth is a spherical body the gravitational field that it produces will have spherical symmetry and we can always consider it to be coming from a point source at ...

Solution Summary

The solution is comprised of an explanation for the simple harmonic motion of the center of the earth. Time to return to point of departure and time needed for a satellite to circle the earth is also considered in the solution.

A hole is drilled straight through thecenter of theearth and a particle is dropped into the hole. Neglect rotational effects:
a) Show that the particle's motion, assuming that earth's gravity acts from the exact center of theearth, is simpleharmonicmotion.
b) Calculate the Period of oscillation

An object executing simpleharmonicmotion has a maximum speed of 4.3 m/s and a maximum acceleration of 0.65 m/s^2.
Find (a) the amplitude and (b) the period of this motion.

An 0.50 kg object is attached to one end of a spring, and the system is set into simpleharmonicmotion. The displacement x of the object as a function of time is shown in the drawing below. With the aid of this data, determine the following values.
(a) amplitude A of themotion
(b) angular frequency
(c) spring c

A body moving with SimpleHarmonicMotion with a frequency of 3Hz has a maximum velocity of 2.5m/s. Find its velocity. Please see attached file for further details regarding problem.
Question - SimpleHarmonicMotion
A body moving with SimpleHarmonicMotion with a frequency of 3Hz has a maximum velocity of 2.5m/s. Find it

A 10-g particle is undergoing simpleharmonicmotion with an amplitude of 2.0 x 10^-3 m and a maximum acceleration of magnitude 8.0 x 10^-3 m/s^2. The phase angle is -pi/3 rad.
a) Write and equation for the force on the particle as a function of time.
b) What is the period of themotion?
c) What is the maximum speed of the p

The pistons in an internal combustion engine undergo a motion that is approximately simpleharmonic. If the amplitude of motion is 3.5 cm, and the engine runs at 1700 rev/min, find (a) the maximum acceleration of the pistons and(b) their maximum speed.

See attached file for full problem description.
1. Consider the four equivalent ways to represent simple harmonic motion in one dimension:
To make sure you understand all of these, show that they are equivalent by proving the following implications: I-->II--> III--> IV. For each form, given an expression for the constants (C

A block with mass attached to a horizontal spring with force constant is moving with simpleharmonicmotion having amplitude. At the instant when the block passes through its equilibrium position, a lump of putty with mass m is dropped vertically onto the block from a very small height and sticks to it.
For this value of m, w