Hooke’s Law Fs = -kx • K is the spring constant, relates to the stiffness of the spring • The negative in front of the k shows that the force is always in the opposite direction as x. • A Slinky has a spring constant of about 1 N/m, a spring in a car’s suspension can have a spring constant of about 1 E 5 N/m
Spring Problems • A 76 N crate is hung from a spring (k = 450 N/m) How much does the spring stretch? • A spring of k = 1962 N/m loses its elasticity if stretched more than .50 m. What is the largest mass that can be hung from this spring without damaging the spring?
Spring Problems • A 76 N crate is hung from a spring (k = 450 N/m) How much does the spring stretch? About 17 cm • A spring of k = 1962 N/m loses its elasticity if stretched more than .50 m. What is the largest mass that can be hung from this spring without damaging the spring? Using hooke’s law, max force is 981 N. Therefore, anything more than about 100 kg would damage the spring.
Oscillations • Springs oscillate in simple harmonic motion • Undamped springs will continue to oscillate • simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.
Oscillations • Damped springs will eventually come to rest • Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations.
Equations for Spring Problems:In absence of external forces such as friction: E= Us + K At max x or equilibrium positions: ½ mv2 = ½ kx2Period of motion:
Equations for Pendulum Problems:In absence of external forces such as friction: E= Ug + K At max h or min h positions: ½ mv2 = mghPeriod of motion:
Oscillations • 1. Equilibrium • x = 0 • F = 0, a = o • v = + max • 2. Max positive displacement • x = + max • F = - max, a = - max • v = 0 • 3. Equilibrium • x = 0 • F = 0, a = 0 • v = - max • 4. Max negative displacement • x = - max • F = + max, a = + max • v = 0 • 5. Equilibrium • x = 0 • F = 0, a = 0 • v = + max
Period vs. Frequency • Period is how long it takes to complete one cycle (seconds per cycle) • Frequency is how many cycles it completes in a unit of time, usually 1 second (cycles per second) • f = 1/T or T = 1/f • SI unit for period – seconds • SI unit for frequency – Hertz, Hz • Hz= 1/s
Period vs. Frequency • A spring makes a complete cycle every 3 seconds, what is its frequency? • A spring has a frequency of 20 Hz, what is its period?
The period of a Spring-Mass System • If you hang a 2kg mass from a spring constant of 25 N/m, what is the period?
A Test-Like Question • At equilibrium 6 N weight stretches a spring 0.5 m. If the spring mass system begins to oscillate, what will be its period? F = kx
Pendulums • Depends on the length of the pendulum and surface gravity. • Pendulums were used as the first way to accurately determine surface gravity
Equations Spring-Mass Pendulum
Problem • How long is the period of a pendulum that is 0.75m long? • If that pendulum is moved to the moon (g = 1.62), what will its new period be?
Periodic Motion vsSimple Harmonic Motion PM- something that repeats the same motion over and over, like the earth going around the sun or a ball bouncing.SHM- a special type of periodic motion that has a restorative force that varies directly with the distance from the equilibrium position or rest position, like a mass on a spring or a pendulum bob.