Using models of SHM and resonance Flashcards
What is the time period for a mass on a spring oscillator?
T = 2pi * (m/spring constant k)^1/2
Outline practical : measuring the period of a mass on a spring.
Attach 4 springs in a row in series to reduce stiffness and increase time period. Raise the spring by displacement x and drop it to count 10 oscillations. Time period squared is proportional to mass, so a graph can be drawn, showing a straight line through the origin.
What is the restoring force for a pendulum and its formula?
When a mass on a pendulum is displaced by x, a restoring force F acts on the ball towards equilibrium, the horizontal component of tension T. F=-Tx/L (L = length of wire)
How is time period found from the restoring force equation for a pendulum?
F=-mgx/L so a=-gx/L=-4pi^2f^2. T=2pi*(L/g)^1/2
What is resonance?
Free oscillations have constant amplitude and vibrate at a particular frequency which is natural. A periodic driving force can cause the system to oscillate at the frequency of the driving force - forced oscillations. Resonance is where natural frequency matches the driving frequency causing large amplitude oscillations.
What is total energy for mechanical oscillators?
Mechanical oscillators store energy, such as a mass on a spring. When displacement is maximum, velocity is zero, so total energy is potential energy, 1/2kx^2. For a pendulum this is GPE. When the velocity is a maximum, total energy is kinetic energy.
Et=Ek+Ep = 1/2 kA^2=1/2m(wA)^2 where w=the angular frequency where A = amplitude.
How does damping affect resonance?
Damping occurs as forces such as frequency cause energy to be taken away. When resonance occurs, the amplitude of oscillations matches until energy losses per cycle = energy supplied. Light damping will cause a dramatic rise in amplitude when freq/natural frequency reaches 1, while heavy damping causes a smaller rise. Beyond the ratio being 1, the amplitude of oscillation will be lower than it was to begin.
How is resonance observed?
Oscillation is driven by a vibrator connected to a signal generator. Amplitude of oscillations measured with a mm scale and a graph of amplitude against frequency is measured.
