Show that the function (sin wt – cos wt) represents simple harmonic motion.
Answers
Answer:
According to question, displacement of a particle in one dimension ( say along x-axis) at time t is given by
x(t)=sinwt−coswt
Taking the derivative of x(t) two times with respect to t, you will get the acceleration
d2xdt2=−w2x
This is the differential equation of motion of a a simple harmonic oscillator.
If the particle has mass m whose displacement is given by x(t) , then multiplying above equation by m , we get the force force on the particle as
F=−Kx
where K=mw2
this is a equation of motion of SHM, whose frequency is w.
(In the above answer, I have assumed that you know calculus! )
OR THIS ALSO COULD BE AN ANSWER
They do not. They provide a visual picture that mimics simple harmonic motion.
They do not. They provide a visual picture that mimics simple harmonic motion.sinθ and cosθ are mathematical functions that define geometrical relationships when a particle moves at a uniform angular velocity along a circle. Such a particle provides a useful mathematical metaphor for simple harmonic process and is a convenient visualization tool.
They do not. They provide a visual picture that mimics simple harmonic motion.sinθ and cosθ are mathematical functions that define geometrical relationships when a particle moves at a uniform angular velocity along a circle. Such a particle provides a useful mathematical metaphor for simple harmonic process and is a convenient visualization tool.Let us obtain the projection of our particle, colored red in the sketch above, onto the y-axis, as it goes round and round along a circular path. The green point we obtain on the y-axis will move up and down, along a straight line, in a simple harmonic manner, in synch with the red particle. The magnitude of oscillation of this green dot at any instant of time is derived from the sine of the angle at which the red particle is located at that time. The number of cycles through which the green particle goes up and down in a second is the ‘oscillation’ frequency of an object undergoing simple harmonic motion.