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Unformatted text preview: r
path of radius r with constant angular velocity ω . Suppose that at time t, the object has swept
out an angle measuring θ radians. If we assume that the object is at the point (r, 0) when t = 0,
the angle θ is in standard position. By deﬁnition, ω = θ which we rewrite as θ = ωt. According
to Theorem 10.3, the location of the object Q(x, y ) on the circle is found using the equations
x = r cos(θ) = r cos(ωt) and y = r sin(θ) = r sin(ωt). Hence, at time t, the object is at the point
(r cos(ωt), r sin(ωt)).11
r Q (x, y ) = (r cos(ωt), r sin(ωt)) 1 θ = ωt
1 rx Equations for Circular Motion
Example 10.2.7. Suppose we are in the situation of Example 10.1.5. Find the equations of motion
of Lakeland Community College as the earth rotates.
Solution. From Example 10.1.5, we take r = 2960 miles and and ω = 12 hours . Hence, the
equations of motion are x = r cos(ωt) = 2960 cos 12 t and y = r sin(ωt) = 2960 sin 12 t , where x
and y are measured in miles and t is measured in hours. In addition to circular motion, Theorem 10.3 is also the key to developing what is usually called
‘right triangle’ trigonometry.12 As we...
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