Simple Harmonic Motion Universal Gravitation

1. Simple Harmonic Motion Vibration about an equilibrium position with a restoring force that is proportional to the displacement from equilibrium. It is a back-and-forth motion over the same path. A force causes the motion to continue to cycle. Equilibrium position is when the object is at rest.

A spring vibrating is SHM. The restoring force is the spring. Pendulum swinging is SHM. The restoring force is gravity.

Velocity is maximum  acceleration is zero  displacement is zero (passing through equilibrium).  balanced forces – inertia keeps it going Velocity is zero  acceleration is maximum (change in direction)  displacement is maximum  Net force is maximum

Simple Harmonic Motion creates waves!

2. Pendulum A mass, called a bob, attached to a string with a pivot point. Swings side to side – simple harmonic motion Restoring force is the gravity. Angle of swing should be 15° or less for SHM. The period is based on the length of the pendulum, not the mass. l = length of pendulum

x Note that the mass of the bob is not a factor in calculating acceleration due to gravity!

3. Mechanical Resonance Application of small forces at regular intervals to a vibrating object causes the size (amplitude) of the vibration to increase. The time interval between applying the force must be the same as the period of oscillation. This is how you keep the swing going! Soldiers do not march across bridges Tacoma Narrows Bridge

4. Universal Gravitation Newton studied the planets and their motion. He proposed a gravitational force. Attractive force that exists between all objects in the universe. Law of Universal Gravitation Inverse square law between F g and d

F g : gravitational force, N G : Universal gravitational constant  G = 6.67 x d : distance between the centers of the masses of the objects. m 1 : mass of one object m 2 : mass of the other object

Henry Cavendish measured the universal gravitational constant by measuring the attraction between spheres. Using this constant, they were able to calculate the mass of the earth.

Calculating the mass of the Earth! F W = m you g F g = m you g =

Objects orbiting the Earth: F c = F g F c = F g

5. Kepler’s Laws Tycho Brahe 1546 – 1601  Believed Earth was the center of the Universe.  Recorded the exact position of the planets and the stars

Johannes Kepler 1571 – 1630  45 years before Newton  Believed in a sun-centered solar system.  Worked as Brahe’s assistant  Used Brahe’s data to formulate the following 3 laws: Kepler’s 1 st Law  Paths of the planets are ellipses, with the sun at one focus.

Kepler’s 2 nd Law  An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals.  Planets move faster when closer to the sun and slower when farther away.

Kepler’s 3 rd Law  The square of the ratio of the periods of any two planets revolving about the sun is equal to the cube of the ratio of their average distances from the sun.   T : period of the planet  r : average distance from the sun

Antares is the 15 th brightest star in the sky. It is more than 1000 light years away!

Example Problems Homework – Universal Gravitation