Simple Harmonic Motion Physics 12

Comprehension Check Determine the orbital radius and speed for satellites with the following periods: 28 days 1 week 1 day 2 hours Determine the orbital speed and period for satellites with the following altitudes: 250km 1000km 6400km

Comprehension Check Determine the orbital radius and speed for satellites with the following periods: 28 days 1 week 1 day 2 hours Determine the orbital speed and period for satellites with the following altitudes: 250km 1000km 6400km

Pendulum We are familiar with a pendulum and have worked with the equation to solve for the period of the pendulum in the lab However, we will go through the concepts of simple harmonic motion in order to develop the equation for the period:

Position of a Pendulum We will consider a simple pendulum that is set into motion and oscillates between –A and A going through 0 Sketch a graph that shows the position of the pendulum as a function of time with it starting at -A

Position of Pendulum

Velocity of a Pendulum Now that we have a graph showing the position as a function of time for a simple pendulum, we wish to consider velocity Sketch a graph that shows the velocity of the pendulum as a function of time with it starting at -A

Velocity of Pendulum

Acceleration of a Pendulum Now that we have a graphs showing the position and velocity as functions of time for a simple pendulum, we will consider acceleration Sketch a graph that shows the acceleration of the pendulum as a function of time with it starting at -A

Acceleration of Pendulum

Pendulum FBD θ mgsin θ mgcos θ

Acceleration of a Pendulum The force accelerating the pendulum in the +/-A position is proportional to the sine of the angle the pendulum makes with the vertical Using this, we are able to see that the acceleration will vary with the position of the pendulum

Acceleration of a Pendulum We have seen that the position varies as a function of time so we use the periodic nature of position We will use A as the amplitude and ω as the angular frequency (where ω=2πf)

Period of a Pendulum According to calculus, the term multiplying the cosine function in the acceleration is equal to the amplitude times the angular frequency squared Further, the term multiplying the sine function in the velocity term is equal to the amplitude times the angular frequency

Mass on a Spring http://vimeo.com/8961180

Acceleration of a Mass on a Spring The force accelerating the mass at the +/-A position is proportional to the amount of stretch and the spring constant Using this, we are able to see that the acceleration will vary with the position of the mass

Acceleration of a Mass on a Spring In a manner similar to how we approached the pendulum, we are able to find the acceleration as a function of time This allows us to write the other equations of motion for the mass on a spring

Period of a Pendulum Once again, using a knowledge of calculus, we are able to show that the angular frequency squared and the angular frequency can both be described using mass and the spring constant This leads to the expression for the period of a mass on the spring

Practice Problems Mass on a spring Pendulum Chapter Review Page 608 Questions 1-4 Pendulum Page 614 Questions 5-8 Chapter Review Page 623 Questions 22-23