Further Mechanics

Angular Speed

The angle an object rotates per second (radians per second).

Frequency (Hz)

Number of complete revolutions per second.

Period

Time taken for a complete revolution (seconds).

Centripetal Acceleration

Acceleration which is always directed towards the center of the circle.

Centripetal Force

Force that always acts towards the center of the circle.

Simple Harmonic Motion of an object

Oscillation of an object from either side of equilibrium. Equilibrium is the midpoint of the object’s motion.

Displacement

Distance of the object from equilibrium.

Restoring Force

Force pulling/pushing the object back towards equilibrium. This makes the object accelerate towards equilibrium.

Simple Harmonic Motion

An oscillation in which the acceleration of an object is directly proportional to it’s displacement (from eq), directed towards eq. So the displacement is the opposite direction of the acceleration.

SHM Graph

• Displacement (Y-axis) varies as a cosine or sine wave with Amplitude as the max/min.

• Velocity is the gradient of the displacement/time graph. Max value of ωA.

• Acceleration is the gradient of the velocity time graph. Max value of ω²A.

When is the acceleration max in SHM?

When there’s zero velocity (which is also when the displacement is at its maximum, negative relative to acceleration).

Phase difference of x, v, a.

Velocity is π/2 radians out of phase (1/4 cycle in front of displacement). Acceleration is another π/2 radians ahead of velocity.

Oscillation (1 Cycle)

Max displacement from right to left to right(+ —> - —> +) or vice versa.

Frequency and period in SHM.

In SHM, frequency and period are independent of the amplitude (constant for given oscillation).

Mechanical Energy

Sum of potential and kinetic energy. This stays constant (as long as the motion is not damped).

Simple Harmonic Oscillator

Mass on a spring: When the mass is pushed/pulled either side of eq, there’s a restoring force exerted on it. (Size and direction of force is found by Hooke’s law).

Free Vibrations

No transfer of energy to or from surroundings. The object will oscillate at its resonant frequency.

Forced Vibrations

Occurs when an external driving force is present. The frequency of said force is called the Driving Frequency.

Driving Frequency < Natural Frequency

Both the driver and oscillator are in phase.

Driving Frequency > Natural Frequency

Oscillator wont be able to keep up, therefore driver is in antiphase with oscillator.