rotational motion uniform circular motion. uniform circular motion riding on a ferris wheel or...
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ROTATIONAL MOTION
Uniform Circular Motion
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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Newton’s 2nd Law• Newton’s 2nd Law
• F = ma
• When objects move in a circle Centripetal acceleration• ac = vt
2 /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
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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
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Centripetal Force
• Centripetal Force• Acts at right angles to an object’s
circular motion• Necessary for circular motion