Pendulums Simple pendulums ignore friction air resistance mass

Pendulums • Simple pendulums ignore friction, air resistance, mass of string • Physical pendulums take into account mass distribution, friction, air resistance • The force that pulls the mass back towards equilibrium is the restoring force

Pendulums • If the restoring force is proportional to the displacement, then the pendulum’s motion is simple harmonic.

Pendulums • For small angles (less than 15°) the pendulum is in simple harmonic motion. • Gravitational PE increases as the displacement increases. Pendulums have gravitational PE and springs have elastic PE. • For pendulums: x↑, PEg ↑ PEg = 0 at equilibrium PE = max; KE = 0 PE = 0; KE = max

Pendulums • The mechanical energy of a simple pendulum is conserved in a frictionless system. • A pendulum’s mechanical energy changes as the pendulum oscillates.

Pendulums • Amplitude = the maximum displacement from equilibrium, measured in radians or meters. • Period (T) = the time it takes for one complete cycle of motion, measured in seconds. • Frequency (f) = the number of cycles or vibrations per unit of time, measured in hertz (Hz). 1 Hz = s-1

Pendulums • Period and frequency are inversely proportional: f = 1/T or T = 1/f

Pendulums • The period of a simple pendulum depends on pendulum length and free-fall acceleration (on Earth it is 9. 81 m/s 2 T = 2π√(L/g) Period = 2π * square root of (length divided by free-fall acceleration)

Pendulums • Shorter pendulums have shorter periods when the acceleration due to gravity is the same. • Mass does not affect the period because while the heavier mas provides a larger restoring force, it also needs a larger force to achieve the same acceleration. Therefore when acceleration due to gravity is the same, pendulums with bobs of different masses (and same length) will have the same period. • Amplitude does not affect the period when the angle is less than 15°.

Springs • But for springs, the heavier the mass on the end, the greater the period: T = 2π√(m/k) Period = 2π * square root of (mass divided by spring constant)

Pendulums Ex: You are designing a pendulum clock to have a period of 1. 0 s. How long should the pendulum be? G: T = 1. 0 s S: 1. 0 s = 2π √(L/9. 81 m/s 2) g = 9. 81 m/s 2 S: 0. 25 m U: L E: T = 2π√(L/g)