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

PE = max; KE = 0

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/s2

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/9.81m/s2) g = 9.81 m/s2 S: 0.25 mU: LE: T = 2π√(L/g)