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Download UNIFORM CIRCULAR MOTION Rotational Motion
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Transcript
ROTATIONAL MOTION Uniform Circular Motion Uniform Circular Motion • Riding on a Ferris wheel or carousel Once a constant rate of rotation is reached (meaning the rider moves in a circle at a constant speed) UNIFORM CIRCULAR MOTION • Recall Distinction: • Speed – • Magnitude or how fast an object moves • Velocity – • Includes both magnitude AND direction • Acceleration – • Change in velocity Preview Kinetic Books- 9.1 Uniform Circular Motion • Uniform Circular Motion • Motion in a circle with constant speed • “Uniform” refers to a constant speed • Velocity is changing though! • Length of the velocity vector does not change (speed stays constant), but the vector’s direction constantly changes • Since acceleration = Change in velocity, the object accelerates as it moves around the track • Instantaneous velocity is always tangent to the circle of motion Uniform Circular Motion • Period • Amount of time to complete one revolution • Period for uniform circular motion • T = 2πr/v (2πr Distance around circle = circumference) • • • • T = period (s) r = radius (m) v = speed (m/s) π = 3.14 Uniform Circular Motion • Tangential speed (vt) • An object’s speed along an imaginary line drawn tangent to the object’s circular path • Depends on the distance from the object to the center of the circular path • Consider a pair of horses side-by-side on a carousel • Each completes one full circle in the same time period but the outside horse covers more distance and therefore has a greater tangential speed Centripetal Acceleration • Centripetal acceleration • Acceleration due to change in direction in circular motion • In uniform circular motion, acceleration = CONSTANT • Points toward the center of the circle perpendicular to the velocity vector • Train goes around a track at a constant speed • Train’s velocity is changing because it is changing direction • Change in velocity = Acceleration Centripetal Acceleration • Centripetal Acceleration • Points toward the center of the circle • ac = vt2 /r • ac = Centripetal acceleration (m/s2) • vt = Tangential speed (m/s) • r = radius of circular path (m) Problem • A car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s2, what is the car’s tangential speed? ac = vt2 / r vt = √acr vt = √(8.05 m/s2)(48.2m) vt = 19.7 m/s Centripetal Force • Forces & Centripetal Acceleration • Yo-yo swings in a circle it accelerates, because its velocity is constantly changing direction • In order to have centripetal acceleration there must be a force present on the Yo-yo • Force that causes centripetal acceleration points in the same direction as the centripetal acceleration Toward the center of the circle Centripetal Force • Any force can be centripetal • Yo-yo moves in a circle by the tension force in the string • Gravitational force keeps satellites in circular orbits • When forces act in this fashion, both tension and gravity Centripetal forces Newton’s nd 2 Law • Newton’s 2nd Law • F = ma • When objects move in a circle Centripetal acceleration • ac = vt2 /r …Now, plug this into F = ma • CENTRIPETAL FORCE (Fc): • Fc = m (vt2/r) • • • • Fc = Newton m = mass (kg) vt = tangential speed (m/s) r = radius of the circular path (m) • Force points toward the center of the circle Problem • A pilot is flying a small plane at 56.6 m/s in a circular path with a radius of 188.5 m. The centripetal force needed to maintain the plane’s circular motion is 1.89 x 104 N. What is the plane’s mass? Fc = mvt2 / r m = Fc r / vt2 = (1.89 x 104 N)(188.5 m)/(56.6 m/s)2 m = 1110 kg Centripetal Force • Centripetal Force • Acts at right angles to an object’s circular motion • Necessary for circular motion