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Illustration 10.1: Coordinates for Circular Motion
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How would you describe the motion of the object shown (position is given in meters and time is given in seconds)? Restart. The object is moving in a circle about x = 0 m and y = 0 m, but the object's x and y coordinates vary with time. They vary in a special way such that x and y are always between -1 m and 1 m. To see this, look at Animation 2 and watch the x and y values change in the table. This is called the component form. We can also describe the motion in terms of the vector form. In this case, the radius vector, r, always has a magnitude of 1 m. but it changes direction. Look at Animation 3. We describe the direction of this vector in terms of the angle it makes with the positive x axis. Therefore the angle-when measured in degrees-varies from 0 to 360. It is often convenient to give the angle in a unit different than degrees. We call this unit a radian. The radian unit is defined as 2π radians = 360°. Notice that both units are defined in terms of one full revolution. To see the angle given in radians look at Animation 4.
So why use radians? Well, it turns out that there is a really nice relationship between angle in radians (θ), the radius (r), and the arc of the circle (s). This geometric relationship states that: θ = s/r. Why is this useful? It allows us to treat circular motion like one-dimensional motion. The arc is the linear distance traveled, which is s = vt when the motion is uniform. This means that θ = (v/r) t, since s = rθ. We call v/r by the name omega, ω, and it is the angular velocity. Therefore, θ = ωt, for motion with a constant angular velocity. When there is a constant angular acceleration, we call it by the name alpha, α, and it is related to the tangential acceleration by at/r. So when we are using radians we can use our one-dimensional kinematics formulas with x → θ, v → ω, and a → α.