Now we are going to bring in the radius r and angular speed omega. A .pdf
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The document describes the motion of a particle moving in a circle of radius 7, with its position given by parametric equations based on time t. It includes calculations for specific time points, angular speed, and linear speed, demonstrating that the particle moves counterclockwise with an angular speed of 3 radians per second and a linear speed of 21 cm/sec. The relation between angular speed and linear speed is also highlighted using the relationship v = omega * r.
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Now we are going to bring in the radius r and angular speed omega. A .pdf
- 1. Now we are going to bring in the radius r and angular speed omega. A particle goes around the circle with radius 7 centered at the origin with the position at time t given by x = 7 cos(3t) y = 7 sin(3t) Mark the points where t = 0, t = pi/4, t = pi/2, t = 3 pi/4, t = pi on the graph. Next to the points, make a note of the t values. Does it go clockwise or counterclockwise? Note that t is not a part of the given graph of y vs. x. How many radians per second does the particle do? What is the speed in centimeters per second? Looking at the previous question, if x = r cos (omega t) y = r sin (omega t). What is the linear speed v in terms of the angular speed omega and the radius r? Solution t =0 , pi/4 , pi/2 , 3pi/4 , pi x = 7cos3t ; y = 7sin3t t = 0 ; x = 7 ; y =0 t = pi/4 ; x= 7cos3pi/4 = -7/sqrt2 ; y = 7sin3pi/4 = 7/sqrt2 t = pi/2 ; x= 7cos3pi/2 = 0 ; y =7sin3pi/2 = -7 t = 3pi/4 ; x = 7cos9pi/4 = 7/sqrt2 ; y = 7sin9pi/4 = 7/sqrt2 t = pi ; x= 7cos3pi =-7 ; y = 7sin3pi =0 In t = pi/4 ; angle through which particle moves =3pi/4 So, angular speed = 3pi/4 / pi/4 = 3 rad/ sec Linear speed = angular speed*radius = 3*7 = 21 cm/sec x = rcoswt ; y = rsinwt v = w*r