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2. A line that connects the planets with the sun sweeps out
equal areas in equal time periods. This means the planet
moves more slowly farther away from the sun and more rapidly
closer to the sun.
3. The square of the period of a planet is directly proportional to
the cube of its orbital radius. (This isn’t a boxer but you will be
asked to derive it)
Note! A uniform shell does not exert a gravitational force on a
particle inside it. As we descend into the earth (or any planet)
only the mass underneath us exerts a net gravitational force.
Ex. With what minimum speed must an object of mass m be launched in order to escape the Earth’s gravitational field?
Ex. A satellite of mass m is in a circular orbit of radius R around the Earth (radius rE, mass M). a) What is the total mechanical energy of the satellite? b) How much work is required to move the satellite into a new orbit, with radius 2R?
Ex. A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 500 N/m. The amplitude of the resulting oscillations is 8.0 cm. Determine the total energy of the oscillator and the speed of the block when it is 4.0 cm from equilibrium.
Ex. A block of mass m = 3.0 kg is attached to an ideal spring of force constant k = 500 N/m. The block is at rest at its equilibrium position. An impulsive force acts on the block, giving it an initial speed of 2.0 m/s. Find the amplitude of the resulting oscillations.
Concept Check. A block is attached to a spring and set into oscillatory motion, and its frequency is measured. If this block were removed and replaced by a a second block with ¼ the mass of the first block, how would the frequency of the oscillations compare to that of the first block?
Concept Check. A student performs an experiment with a spring-block simple harmonic motion oscillator. In the first trial, the amplitude of the oscillations is 3.0 cm, while in the second trial, the amplitude of the oscillations is 6.0 cm. Compare the values of the period, frequency and maximum speed of the block for the two trials.
Ex. A block of mass m = 1.5 kg is attached to the end of a vertical spring of force constant k = 300 N/m. After the block comes to rest, it is pulled down a distance of 2.0 cm and released. a) What is the frequency of the resulting oscillations? b) What are the minimum and maximum distances the spring stretches during the oscillations of the block?
Ex. A simple harmonic oscillator has an amplitude of 3.0 cm and a frequency of 4.0 Hz. At time t = 0, its position is x = 0. Where is it located at time t = 0.30 s?
Differential equation for SHM. Any object that has motion
described in this form undergoes SHM. Or, the acceleration of any
object that undergoes SHM motion is described by this
equation. As every third grader knows, this differential equation
is the hallmark of SHM.