Question

A particle travels counterclockwise around the origin at constant speed in a circular path that has a diameter of 2.50 m, going around twice per second. Point Q is on the path halfway between points P and R, which lie on the x axis and the y axis, respectively. State the following velocites as vectors in polar notation. (a) What is the particle's average velocity over the interval PR? (b) What is its average velocity over the interval PQ? (c) What is its instantaneous velocity at point P?

P coordinates: (1.25, 0)
R coordinates: (0, 1.25)
Q is directly between the two

Answer 1

frequency f = 2 rev/s
angular velocity = 2•π•f = 4•π rad/s
(a) Velocity vector at P = (1.25x4x3.14)j = 15.7 j,
where j is unit vector along positive y axis.
Also velocity vector of the particle at R = -15.7i,
where i is unit vector along positive x direction.
As the direction changes uniformly,
the average velocity over the interval PR is
(15.7/2)(j-i)= 7.85(j-i).
Magnitude = sqrt[7.85^2+7.85^2] =
=1.414x7.85 = 11.10 m/s
and the direction is 45 degrees west of north.
(b) Average velocity, V1 over interval PQ is
= 0.5{15.7 j + [15.7/sqrt(2)](j-i)} or
V1 = 0.5[26.80 j - 11.10i] = 13.40 j - 5.55 i.
Magnitude of V1 = sqrt[13.40^2 + 5.55^2] = 14.50 m/s.
due theta west of north given by theta = tan^-1 [5.55/13.40] = 22.5 degrees west of north.
(c) v= 2•π•f•R = 15.7 m/s due north