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Further Mechanics (SHM (Velocity (\(\dot{x}=A\,\omega\cos{\omega\,t}\)
or…
Further Mechanics
SHM
When the acceleration of a particle is proportional to the distance of the particle from an equilibrium position, and the acceleration is directed towards equilibrium
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Displacement
\(x=A\sin{\omega\,t}\)
or
\(x=A\cos{\omega\,t}\) #
Velocity
\(\dot{x}=A\,\omega\cos{\omega\,t}\)
or
\(\dot{x}=-A\,\omega\sin{\omega\,t}\) #
\(\dot{x}=\pm\,\omega\sqrt{A^{2}-x^{2}}\)
\(\dot{x}_{max}=\omega\,A\)
Acceleration
\(\ddot{x}=-A\,\omega^{2}\sin{\omega\,t}\)
or
\(\ddot{x}=-A\,\omega^{2}\cos{\omega\,t}\)
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Energy
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At maximum and minimum displacement potential energy is a maximum and kinetic energy is zero (relative to the oscillating system)
Driven Harmonic Motion
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Resonance peak
Graph of resultant amplitude of oscillation against driving frequency, when damping present
Starts at (0, driving amplitude)
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Resonance
When driving frequency equals the natural frequency of the oscillator, and so the amplitude of oscillation becomes very large
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Damping
When energy is gradually removed from an oscillating system by action of a force that acts in the opposite direction to the velocity of the oscillator, and is proportional to the magnitude of the velocity.
Critical Damping
Damping which removes all the energy, and therefore stops the oscillations, in the shortest time possible. There is no oscillation around equilibrium with critical damping
Over Damping
Damping that is heavier than critical damping, such that it takes even longer for the oscillating system to come to rest.
Light and Heavy Damping
Damping that removes energy slower and faster respectively. An underdamped oscillator will oscillate around equilibrium before coming to rest at equilibrium
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Circular Motion
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Angular Velocity \(\omega\)
The angle through which an object rotates through per second
\(rad\,s^{-1}\) \[\omega=\frac{\theta}{t}\]
Tangential Velocity \(v\)
The linear velocity of a circularly moving object at a particular instant in time
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Free vibrations involve no transfer of energy to or from the oscillating system from or to it's surroundings