Suppose that a particle vibrates in such a way that its position function is r(t) = 16 sinπti + 4cos 2πtj, where distance is in millimeters and t is in seconds. (a) Find the velocity and acceleration at time t = 1s. (b) Show that the particle moves along a parabolic curve. (c) Show that the particle moves back and forth along the curve.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
Suppose that a particle vibrates in such a way that its position function is r(t) = 16 sinπti + 4cos 2πtj, where distance is in millimeters and t is in seconds.
(a) Find the velocity and acceleration at time t = 1s.
(b) Show that the particle moves along a parabolic curve.
(c) Show that the particle moves back and forth along the curve.
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