Question

A particle undergoes uniform circular motion. This means that it moves in a circle of radius R about the origin at a constant speed. The position vector of this motion can be written Here, analogous to the simple harmonic motion problem of HW 1, ω is the angular frequency and has units of rad/s 1/s and can also be written in terms of the period of the motion as 2π (a) Show that the particle resides a distance R away from the origin at all times. That is, show that the magnitude of F(t) is R for all times. (b) Calculate the velocity vector of this motion. How is the velocity vector oriented relative to the position vector? For example, is it parallel, perpendicular or at a particular angle relative to the position vector? (c) Calculate the speed of the particle for all times. Show that the speed can also be written as (d) Calculate the acceleration vector for this motion. How is the acceleration vector oriented relative to the position vector? For example, is it parallel, perpendicular or at a particular angle relative to the position vector? (e) What is the magnitude of the acceleration vector for all times? Show that it can be written as

Answer 1

r^rightarrow (+) = R cos wt i^+ R sin wt j^ Vertical-bar r^rightarrow Vertical-bar = squareroot (R cos wt)^2 + (R sin wj)^2 = squareroot R^2 cos^2 wt + R^2 sin^2 wt = squareroot R^2 [sin^2 wt + cos^2 wt] {we know sin^2 theta + cos^2 theta = 1} therefore Vertical-bar r^rightarrow Vertical-bar = squareroot R^2 = R rightdoublearrow Vertical-bar r^rightarrow Vertical-bar = R (1) magnitude is indefind + of times. B) V^rightarrow = d r^rightarrow/dt = -R sin wt (w) i^+ R (cos wt) w j^{d/d theta sin theta = cos theta, d/d theta cos theta = -sin theta} V^rightarrow = -R w sin wt i^+ R w cos wt j^(II) c) speed = Vertical-bar V^rightarrow Vertical-bar = squareroot (- R w sin wt)^2 + (R w cos wt)^2 = (R w) squareroot sin^2 wt + cos^2 wt {substitute w = 2 pi/T}Vertical-bar V^rightarrow Vertical-bar = 2 pi R/T (3) d) a^rightarrow = d v^rightarrow/dt = -R w^2 cos wt i^- R w^2 sin wt j^(4)