Physics C.1 Simple harmonic motion

Cycle

One cycle is defined as one complete oscillation of a pendulum. The term cycle is also used to describe circular motion; one cycle is one complete circle or 2pi radians.

Equilibrium position

Where the pendulum bob would rest if not disturbed

Amplitude (x₀)

Maximum displacement from the equilibrium position.

Time period (T)

Time taken for one complete cycle

Frequency (f)

number of cycles that the pendulum makes per unit time. This is equal to 1/time period.

Angular frequency (w)

Found by multiplying f by 2pi. An angular frequency of 2pi rads^-1 means that a body makes one revolution per second or one complete cycle for oscillation.

Simple harmonic motion

Defined as the motion that takes place when the acceleration of an object is always directed towards the equilibrium position and is proportional to the displacement from a central position.

Resultant forces

The weight is constant but the resultant between the weight and tension changes. The resultant increases as the amplitude increases. Acceleration is proportional to displacement -x.

Approximations of simple harmonic motion (when the angle is very small)

1. the displacement is horizontal

2. The force acting towards the equilibrium position is the horizontal component of the tension, F_tsinx (in reality, it is the resultant of the weight and the tension)

3. The weight is approximately the same as the tension (in reality, the tension is greatest at the bottom of the swing) F_t=mg so the restoring force = F_tsinx=mgsinx

Displacement - time graph SHM

• Sinusoidal

• y=Asinx

Mass on a spring

• forces acting on the mass are the weight and tension

• weight is always the same but the tension depends on how far the spring is stretched

• acceleration is always proportion to the displacement

• so it is simple harmonic motion

Velocity in terms of displacement

• max velocity is at min displacement and min acceleration

• min velocity is at max displacement and max acceleration

Kinetic energy in SHM

Maximum kinetic energy is at equilibrium (=total energy and potential energy = 0)

Potential energy in SHM

Maximum potential energy is at the amplitude (=total energy and kinetic energy = 0)

Total energy

Since no work is done on the system, according to the law of conservation of energy, the total energy must be constant. If the graphs of potential energy and kinetic energy are added together, it gives a constant value, equal to the total energy.

Phase

In phase: same displacement at the same time of two identical pendulum bobs

Out of phase: one is pulled to the left and the other to the right

Phase difference

Difference in phase between two points. Represented by an angle (usually in radians). If two oscillations are completely out of phase, then the graphs are displaced by an angle of pi. So the phase difference is pi.