To find the velocity and acceleration vectors for uniform circular motion and to
ID: 1531513 • Letter: T
Question
To find the velocity and acceleration vectors for uniform circular motion and to recognize that this acceleration is the centripetal acceleration. Suppose that a particle's position is given by the following expression T(t) = R [cos(omega t)i + sin(omega t)j] = R cos (omega t)i + R sin (wt)j What type of motion does this represent? Using dimensional analysis, what units does omega have? Find the particle's velocity as a function of time Find the particle's acceleration as a function of time. What is the magnitude of the velocity? Express your answer in terms of omega and R. What is the magnitude of the acceleration? Express your answer in terms of omega and R Express the magnitude of the acceleration in terms of v and R.Explanation / Answer
Given,
r(t) = R cos(wt)i + R sin(wt) j
a)The motion is circular.
b)units of w should be rad/s.
c)we know from defination of velocity
v(t) = dr(t)/dt
v(t) = d (R cos(wt)i + R sin(wt) j)/dt
v(t) = - R w sin(wt) i + R w cos(wt) j
Hence, v(t) = R w [ -sin(wt)i + cos(wt)j ]
d)a(t) = dv(t)/dt
a(t) = d [R w [ -sin(wt)i + cos(wt)j ]]/dt
a(t) = R w^2 [ -cos(wt)i - sin(wt)j]
a(t) = - R w^2 [ cos(wt)i + sin(wt)j]