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PHYS_2014_Lecture_7 - Lecture 7 Circular Motion v r I swing...

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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 1 r v I swing a ball attached to a string of length r over my head. The path that the ball takes is a circle of radius with my hand at the origin. At any given instant, the velocity, , of the ball is a vector tangent to the circular path, i.e. . arrowrightnosp v arrowrightnosp r arrowrightnosp arrowrightnosp v r Lecture 7: Circular Motion
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 2 Circular or rotational motion , while common in everyday life, is different from the linear motion we have studied so far. Circular motion is clearly different from the straight-line motion of objects sliding or rolling over surfaces (inclined or otherwise) and is clearly different from projectile motion wherein objects follow parabolic trajectories. One difference is that circular motion repeats...the object traverses the same path, i.e. the same circular trajectory, over and over again. r v
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 3 Because the motion is constrained to a circle, we can specify all possible positions where the object (ball) might be in terms of the radius, , and the angle, θ . Thus, it makes sense to use a polar coordinate system. 2 2 1 ˆ cos ˆ sin tan = = = + = arrowrightnosp arrowrightnosp x r i y r j r x y y x θ θ θ θ r y x arrowrightnosp r
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 4 r Δ θ s x The distance traveled by an object undergoing circular motion can be found by the radius r and the change in the angle Δ θ . (radians) s r θ = Δ (radians) s r θ Δ = s is not in a straightline . It is the segment of the circumference of a circle and is referred to as arc length.
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 5 The length of one circular path around my hand is the circumference of the circle of radius r . If the ball I’m swinging above my head takes a time T (the period) to complete one circular path, the the ball’s speed will be: 2 s r π = 2 s r v t T π Δ = = Δ r v
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 6 We can apply this to the Earth’s motion around the Sun. The average distance between the Sun and the Earth is 1.49 × 10 11 m and the time it takes the Earth to go around (orbit) the Sun one time is 365.25 days or 3.16 × 10 7 s. 11 4 7 2 1.49 10 m 2.91 10 m s 65,000miles hr 3.16 10 s v π × = = × ×
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Fall 2010 Oklahoma State University PHYS2014: Benton Lecture 7, Slide 7 Consider our original definitions for velocity and acceleration and s v v a t t Δ Δ = = Δ Δ arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp As defined here, position, velocity, and acceleration are all vectors. By definition a vector has both magnitude and direction.
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