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Circular Motion

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Circular Motion: Description & Causes Circular motion is motion along a circular path due to the influence of a centripetal force. [Note: Centripetal means “center seeking.”] Circular motion is caused by a centripetal force F c which is any net force of constant magnitude always pointing toward a single point, the center of a circle. Centripetal acceleration a c is the “center seeking” acceleration caused by a centripetal force.

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Centripetal acceleration Centripetal acceleration a c is the acceleration resulting from the change of the direction of an object with no change in speed. The speed is constant but the velocity is not because direction is always changing. Centripetal acceleration also points toward the center of a circle just like centripetal force does. In fact, both centripetal force and centripetal acceleration point parallel to the radius toward the center.

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Tangential speed (“how fast”) Tangential speed is the constant speed at which a body in circular motion moves. It is sometimes identified as the linear speed because that’s how fast the object would move in a straight line if there were suddenly no centripetal force acting on it. Tangential velocity has constant magnitude but changing direction. Its vector arrow points perpendicular to the radius, acceleration, and force.

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vTvT acac FcFc Diagram of an object moving clockwise Note the direction and orientation of the centripetal force, centripetal acceleration, and tangential velocity. Dashed line is the path of the object if there is no centripetal force.

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Terms, Symbols, & Units Physical Terms Equation Symbol SI Unit Tangential Speed v t m/s Centripetal Acceleration a c m/s 2 Centripetal Force F c N Radius r m

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Equations for Circular Motion F c = m∙a c F c is directly related to m & a c a c = v t 2 /r a c is related to the square of v t F c = m∙v t 2 /r F c and a c are inversely related to r

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Example Problem #1 A 10 kg block of wood is whirled around on a rope with an acceleration of 5 m/s 2. How much centripetal force acts on it to keep it moving in a circle? Solution: F c = ma c = (10 kg)(5 m/s 2 ) F c = 50 N

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Example Problem #2 At what speed must a block of wood move at the end of a 1 m length of rope to maintain a 25 m/s 2 centripetal acceleration? Solution: a c = v t 2 /r 25 m/s 2 = v t 2 / (1 m) v t 2 = 25 so….. v t = 5 m/s

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Example Problem #3 A 5 kg block of wood revolves around at 4 m/s maintained by a centripetal force of 20 N. At what radius does it revolve? [What’s the length of rope to which it’s attached?] Solution: Fc = mv 2 /r 40 N =(5 kg)(4 m/s) 2 /r 40 = 80/r r = 80/40 = 2 m

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Proportion Problems The original centripetal force acting on a block of wood revolving in a circle is 10 N. F new = (multiplication factor) x F orig Doubling the mass of the block will double the force needed to keep it moving in a circle, so… F new = 2 x 10 N = 20 N Doubling the speed will quadruple the force needed to maintain a circle so F new = 2 2 x 10 N = 40 N Doubling the radius will change the force by ½ so F new = ½ x 10 N = 5 N

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Terminology & Misc. Concepts Rotation is circular motion about an internal axis such as the rotation of the Earth causing night and day. Revolution is circular motion about an external axis such as the Earth moving around the Sun or the Moon around the Earth. Time for one revolution T= Circum./Speed = (2πr/v) so a path with a greater radius (r) and greater circumference (2πr) requires a faster speed to complete one trip in the same time.

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