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Hamiltonian
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Under the topic of simple harmonic motion comes the composition of two SHM's with the same angular frequency, different phase constants, and amplitudes in the same directions and in perpendicular directions.
composition of SHM's in same direction:
say a particle undergoes two SHM's described by the following equations:
$$x_1 = A_1sin(\omega t)$$
$$x_2 = A_2sin(\omega t + \phi)$$
then the resulting SHM is described by the equation:
$$x = x_1 + x_2 = Asin(\omega t + \theta)$$
we can find the values of ##A##(amplitude of resulting SHM) and ##\theta##(phase constant of resulting SHM) quite easily.
I wanted to know if there were any real-life(by "real-life" I simply mean something with springs and blocks and wedges and what not) examples where these equations could be applied instead of just a hypothetical particle undergoing 2 hypothetical SHM's. the first thing that came to my mind was something like this
(assume the two springs have a different value of spring constant(k) say ##k1## and ##k2##)
then the equations of motion are:
$$x_1 = A_1 sin(\sqrt{\frac{k_1}{m}} t)$$
$$x_2 = A_2 sin(\sqrt{\frac{k_2}{m}}t)$$
but I am unsure as two how to (or if I even can) proceed to prove that this the composition of SHM's in the same direction. from the above equations of composition of SHM's the resulting amplitude of the motion should be ##A_1+ A_2##
I didn't have any problem with the composition of SHM's in the perpendicular direction.composition of two SHM's
composition of SHM's in same direction:
say a particle undergoes two SHM's described by the following equations:
$$x_1 = A_1sin(\omega t)$$
$$x_2 = A_2sin(\omega t + \phi)$$
then the resulting SHM is described by the equation:
$$x = x_1 + x_2 = Asin(\omega t + \theta)$$
we can find the values of ##A##(amplitude of resulting SHM) and ##\theta##(phase constant of resulting SHM) quite easily.
I wanted to know if there were any real-life(by "real-life" I simply mean something with springs and blocks and wedges and what not) examples where these equations could be applied instead of just a hypothetical particle undergoing 2 hypothetical SHM's. the first thing that came to my mind was something like this
then the equations of motion are:
$$x_1 = A_1 sin(\sqrt{\frac{k_1}{m}} t)$$
$$x_2 = A_2 sin(\sqrt{\frac{k_2}{m}}t)$$
but I am unsure as two how to (or if I even can) proceed to prove that this the composition of SHM's in the same direction. from the above equations of composition of SHM's the resulting amplitude of the motion should be ##A_1+ A_2##
I didn't have any problem with the composition of SHM's in the perpendicular direction.composition of two SHM's
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