Question

An object's position is given by the vector:
s(t)=2cos(wt)i^+2sin(wt)j^

Where w=1 radian per second.
How would you describe this object's motion?

Answer 1

s(t) = 2 cos wt i + 2 sin wt j

w = 1 rad/s

s(t) = 2 cos t i + 2 sin t j

The x and y components of the displacement vector s(t):

x(t) = 2 cos t

In general, the cartesian coordinates x, and y of a particle motion can be related to its radial coordiantes r and theta as:

x = r cos (theta)

and y = r sin (theta)

where r is the radial coordinate and theta is the angular coordinate

and the vector can be written as:

A = r cos (theta) i + r sin (theta) j

Compare with what we have:

s(t) = 2 cos wt i + 2 sin wt j

We see, r = 2, theta = wt = t (w=1)

Thus we can say that the particle is moving in a circle with radius = 2 units and theta = wt represents angular position with respect to x axis.

w is the angular frequency of circular motion of the particle = 1 radians per sec

So the particle completes 1 radian in 1 sec

or, it completes one full revolution in 2*pi = 2*3.14=6.28 seconds.

More specifically, at t = 0 sec, x = 2 cos 0 = 2 units on x axis, y = 2 sin 0 =0 units

thus particle is on x axis = 2 units

at t = pi/2 = 3.14/2 seconds, it is at:

x = 2 cos pi/2 = 0 units

y = 2 sin pi/2 = 2 units

or the particle is on y axis at y = 2 units and x =0

at t = pi = 3.14 seconds, x = -2, y = 0

at t = 2*pi = 6.28 it comes back to its starting point : x=2, y=0.