Oscillations

Definitions Displacement is measured from the equilibrium point. Amplitude is the maximum displacement. A cycle is a full to-and-fro motion. Period is the time required to complete one cycle. Frequency is the number of cycles completed per second. 𝑓= 1 𝑇

Frequency Measured in Hertz (Hz) 1 𝐻𝑧=1/𝑠

Oscillations of a Spring If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.

Effect of mass and the force constant on the period For a spring that exerts a linear restoring force the period of a mass-spring oscillator increases with mass and decreases with spring stiffness.

Period and frequency of mass-spring system with SHM 𝑇=2𝜋 𝑚 𝑘 since f = 1/T, 𝑓= 1 2𝜋 𝑘 𝑚 𝜔=2𝜋𝑓= 𝑘 𝑚

Oscillations of a Spring

Oscillations of a Spring We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure). The force exerted by the spring depends on the displacement: 𝐹 𝑠 =𝑘 𝑥

Oscillations of a Spring The restoring force is directed opposite to the direction of motion of the mass to restore the mass to its equilibrium position. k is the spring constant. The force is not constant, so the acceleration is not constant either.

Oscillations of a Spring If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

Simple Harmonic Motion Any motion that repeats itself is periodic or harmonic. If the motion is a sinusoidal function of time, it is called simple harmonic motion (SHM). Simple harmonic motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Position as a function of time x = A cos (t) Where: A = the amplitude (maximum displacement of the system) t = time w = angular frequency  = 2f

Velocity and Acceleration as a function of time 𝑣=−𝜔𝐴 sin 𝜔𝑡 𝑎=− 𝜔 2 𝐴 cos 𝜔𝑡

Maximum velocity In simple harmonic motion, the velocity constantly changes, oscillating just as the displacement does. When the displacement is maximum, however, the velocity is zero; when the displacement is zero, the velocity is maximum. Since the velocity is given by: 𝑣=−𝜔𝐴 sin 𝜔𝑡 So the maximum velocity is: 𝑣 𝑚𝑎𝑥 =𝐴𝜔

Maximum acceleration 𝑎=− 𝜔 2 𝐴 cos 𝜔𝑡 If you consider a mass on a spring, when the displacement is zero the acceleration is also zero, because the spring applies no force. When the displacement is maximum, the acceleration is maximum, because the spring applies maximum force; the force applied by the spring is in the opposite direction as the displacement. Since acceleration is given by: 𝑎=− 𝜔 2 𝐴 cos 𝜔𝑡 So the maximum acceleration is 𝑎 𝑚𝑎𝑥 = 𝐴𝜔 2

Velocity as a function of position Using the conservation of energy, 𝐸 𝑡𝑜𝑡𝑎𝑙 =𝐾+ 𝑈 𝑠 1 2 𝑘 𝐴 2 = 1 2 𝑚 𝑣 2 + 1 2 𝑘 𝑥 2 𝒗= ± 𝒌 𝒎 ( 𝑨 𝟐 − 𝒙 𝟐 ) Note: speed is maximum at x = 0 and zero at x = A. Note: 𝑬 𝒕𝒐𝒕𝒂𝒍 = 𝟏 𝟐 𝒌 𝑨 𝟐

Acceleration 𝒂= 𝒌 𝒎 𝒙 Using Newton’s second law: F = ma kx = ma 𝒂= 𝒌 𝒎 𝒙 Note: acceleration is maximum when x = A

Maximum velocity 𝑣 𝑚𝑎𝑥 =2𝜋𝑓𝐴

a is the angular acceleration of the mass. Finally, Pendulums In a simple pendulum, a particle of mass m is suspended from one end of an unstretchable massless string of length L that is fixed at the other end. The restoring torque acting on the mass when its angular displacement is q, is: a is the angular acceleration of the mass. Finally, This is true for small angular displacements, q.

Effect of length on the period of the pendulum For a simple pendulum oscillating the period increases with the length of the pendulum.

 (degrees)  (radians) sin  Pendulums In the small-angle approximation we can assume that  << 1 and use the approximation sin   . Let us investigate up to what angle  is the approximation reasonably accurate?  (degrees)  (radians) sin  5 0.087 0.087 10 0.174 0.174 15 0.262 0.259 (1% off) 20 0.349 0.342 (2% off) Conclusion: If we keep  < 10 ° we make less than 1 % error.