AP Physics C I.F Oscillations and Gravitation

Kepler’s Three Laws for Planetary Motion

1. All planets move in an elliptical orbit (barely) with the sun at one of the foci.

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.

Conservation of angular momentum revisted

Newton’s Universal Law of Gravity The falling apple

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)

Gravitational attraction due to an extended body

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.

The mass falling from pole to pole

The speed of a satellite

Gravitational Potential Energy

Escape velocity

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 r E, 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?

Oscillations

A block on a spring

Note: the net force is zero and the speed is maximum when the block is at its equilibrium position

A quick review on energy and SHM

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.

The spring-block oscillator for vertical motion

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?

Time out for a calculus lesson

Instantaneous velocity and accleration

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.

Period of a spring revisited

Describing the motion of a simple pendulum

Ex. A meter stick swings about a pivot point a distance L from its center of mass. What is its period?