The position of a five gram mass is given by r(t) = Asin(3t) i-hat + Bcos(2t) j-

Question

The position of a five gram mass is given by r(t) = Asin(3t) i-hat + Bcos(2t) j-hat + C k-hat A,B,C are constant lengths, i-hat is the unit vector in the x-direction, etc. In terms of A,B, and C, How far from the origin is the particle at t = 0? What is its velocity at t=0? What is the force on it at t=0? What is the force on it at t=pi/4? How long does it take for the x-coordinate to complete one full cycle? How long for the y-coordinate? This is a two-dimensional oscillator, with different angular frequencies in the x- and y-directions.For fun (yes, this is fun), plot this out up to t= 2n or more. Let C=O; choose any values for A and B. Better yet, write a program to plot it out. Can you see that the system repeats itself every 2pi seconds? (It returns to its starting position in pi seconds, but it is traveling in the opposite direction compared to t=0.)

Explanation / Answer

At t=0, particles at r= B j + C k or (b^2 + c^2) from the origin.

differentiating wrt t

Vx = 3Acos(3t) Vy = -2Bsin(2t) Vz = 0

Thus t=0 , velocity = 3A i m/s

Again differentiating wrt t,

Ax = -9Asin(3t) Ay = -4Bcos(2t) Az = 0

Acc at t=0 is -4B j m/s^2 Force at t=0 = 0.005 * -4B j Newton = -0.02B Newtons

Acc at t=pi/4 is (-9A/2) m/s^2 Force at t=pi/4 is -0.045A/2 newtons

Along x, =3 => T =2*pi/ = 2*pi/3. (ans)

along y, = 2 =>  T =2*pi/ = 2*pi/2. (ans) = pi

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