Describing Periodic Motion AP Physics

Hooke’s Law

Restoring Force The force exerted by a spring is a restoring force: it always opposes any displacement from equilibrium

Elastic Potential Energy Work done is the area under the force vs. displacement graph The area in this case can be found without calculus

Elastic Potential Energy

Periodic Motion Any motion which repeats itself is periodic. The time it takes to compete a cycle is the period of the system. Examples: Perfect Bouncy Ball, Pendulum, Mass on a spring, spinning object Example: Mass on SpringMass on Spring

Harmonic Motion If a linear restoring force restrains the motion of an object, then the periodic motion is called simple harmonic motion The system is called a Simple Harmonic Oscillator (SHO)

Harmonic Motion Harmonic motion can be mathematically described by a sine function.

Energy Conservation If no energy is lost, a mass on a spring will remain in motion forever. Sacred Tenant of Physics: The total energy of the system will be conserved!

Energy Conservation

Example A 1 kg. mass is attached to 25 N/m spring, stretched 10 cm from equilibrium and then released. What is the energy stored in the system before being released? What is the maximum velocity of the mass? What is the velocity when the mass is at x=5 cm?

Circular Motion Simple Harmonic Motion can be compared with circular motion. Demo Derive the period of the system

Finding the Period

Period and Frequency

Angular Frequency

Mathematical Model

Example 2 Write an equation for the position of a 0.3 kg. mass on a 100 N/m spring that is stretched from it’s equilibrium position of 15 cm to 18 cm then released. Find the period of the system, T Determine the angular frequency,  Determine the Amplitude, A x(t) = Acos(  t)+x o.

Example 3 The position function of a 100 g. mass is given by Determine the following:

Example 3 Solutions