oscillationssimple harmonic motion: In oscillations, we observe that the motion is repetitive about a fixed point, with the object at rest at the extremes of the motion and moving with maximum speed in either direction at the midpoint. Amplitude is the maximum displacement from the mean position. Period is the time taken to complete on oscillation. Frequency is the complete oscillations per second. An oscillation that satisfies these conditions is said to have a simple harmonic motion: The force acting on the oscillating body, and therefore its acceleration, is proportional to the displacement of the body. The force, and therefore the acceleration, always acts in a direction towards the equilibrium position. F = -k*x The minus sign means the force is always opposite in direction to displacement. A = -w2 * x, where w2 is the constant by convention.Spring: A = -w2 * x Oscillation period is given by: T = 2 * * root(m/k)simple pendulum: In case of a single pendulum the force causing oscillation is provided by a component of the weight. The required force is the horizontal component of weight (the one with sin). T = 2 * * root (L/g), where L is the distance from the point of suspension to the center of gravity.Equations of simple harmonic motion: In perfect simple harmonic motion the frequency of the oscillations does not depend on the amplitude of oscillation. We say the motion is isochronous. X = A * cos(wt) The equations for acceleration and velocity are found by differentiating the above equation. V = -w * A * sin(wt) The acceleration is given by the gradient of the velocity-time graph at any instant. a = -w2 * A * cos(wt) w = 2 * * fEnergy in simple harmonic motion: As a pendulum swings to and fro there is a continuous interchange of kinetic and potential energy. Energy of a system of pendulum = 0.5 * m * w2 * A2 A free oscillation is one in which no external force acts on the oscillating system except, of course, the force causing the oscillation. An oscillating system does work against the external forces acting on it, such as air resistance, and so uses up some of its energy. This transfer of energy from the oscillating system to internal energy of the surrounding air causes oscillations to slow down and eventually die away-the oscillations are damped. Forced oscillation is felt when the receiver and source have the same vibrating frequency.Resonance: Pendulum X oscillates at its natural frequency. All other, light pendulums experience forced oscillations, equal in frequency to that of the driving pendulum. The pendulums of length nearest to the length of the driving pendulum will absorb more energy because their natural frequency of vibration is close to that of the driving pendulum. These pendulums oscillate with larger amplitudes. A pendulum having the same natural frequency (same length) as the driving pendulum will absorb by far the most energy and will be forced to oscillate with very large amplitude. This is called resonance. For a system with little or no damping, resonance occurs when the applied frequency equals the natural frequency. If there is damping, then the resonant frequency at which the amplitude is a maximum is lower than the natural frequency, and that this difference increases as the degree of damping increases. However, the maximum energy transfer, or energy resonance, always occurs at the natural frequency. As the amount of damping increases, the resonant peak is much lower and the resonance curve broadens out. Damping is important in the design of machines and buildings to prevent unwanted vibrations, which if they built up to large amplitude through resonance could cause severe damage. Various alterations in design are used to tackle the problem. The shape of a lathe is designed so that its resonant frequency is nowhere near the frequency of rotation of the lathe, and a car engine is mounted on special dampers to absorb the energy of the vibrations. If vibrations occur in a ductile material, the material goes through a hysteresis loop each vibration. This absorbs the energy and prevents vibrations of large amplitude building up.