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Unformatted text preview: Ch. 8 Rotational Kinematics 8.1 Rotational Motion Ch. 2 focused on Kinematics in 1D, describing the motion of an object as it moves linearly – along a straight line. In Ch. 5 we studied uniform circular motion, where an object moves along a circular path at constant speed. Now let’s analyze objects moving on a circular path, where the speed can be changing, thus the object can be accelerating. What is a circle??? A circle is a locus of points in a plane equidistant from a fixed point in the plane. In rotational motion, the center of the circular motion becomes the axis of rotation. In Ch. 2 we defined the change in linear displacement as: o f x x x = ∆ We can define the change in the angular displacement as: o f θ θ θ = ∆ *This is the angle swept out by a rigid body rotating in a plane about a fixed axis of rotation which is perpendicular to that plane. The “vector nature” of the angular displacement is associated with the direction of rotation : ∆ θ is positive for counterclockwise (ccw) rotations. ∆ θ is negative for clockwise (cw) rotations. Units? Radians [rad] 1 revolution = 2 π rad We could also use degrees: 1 revolution = 360 o 2 π rad = 360 o As an object rotates, it traces out an arc of length s , where: θ r s = Rearrange the equation for arc length: r s = θ Notice, θ must be unitless! Radians have no effect on other units. Look at two circles with different radii from above: r 1 r 2 s 2 s 1 2 π 2 π o θ o θ f θ f θ In both case the angular displacement ∆ θ = π /2, but the arc lengths are different, since the radii are different. Example : A space station consists of two donutshaped living chambers, A and B, that have the radii shown in the figure. As the station rotates, an astronaut in chamber A is moved 2.40×10 2 m along a circular arc. How far along a circular arc is an astronaut in chamber B moved during the same time? From above: A B A s B s θ θ is the same for both chambers. We can find it from the arc length for chamber A: θ A A r s = A A r s = ⇒ θ rad 75 . m 10 2 . 3 m 10 4 . 2 2 2 = × × = θ B B r s = m 825 rad) m)(0.75 10 1 . 1 ( 3 = × = 8.2 Angular Velocity and Acceleration In Ch. 2 we defined velocity as in time Change nt displaceme in Change We can do the same thing for rotational motion: in time Change nt displaceme angular in Change locity angular ve Average = t ∆ ∆ = θ ϖ Units?...
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 Spring '08
 SPRUNGER
 Physics, Circular Motion, Rotation, Angular Acceleration

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