rotational mechanics. rotary motion rotation about internal axis (spinning) rate of rotation can be...
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Rotational MechanicsRotational Mechanics
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Rotary MotionRotary Motion
Rotation about internal axis (spinning)Rate of rotation can be constant or variableUse angular variables to describe rotationAll parts of a rigid rotating object have same angular displacement, velocity and acceleration Linear displacement and velocity depend on distance from rotation axis.
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Radian Measure of AnglesRadian Measure of Angles
Radian is ratio of arc distance subtended by angle to the radius: = darc/r
Radian is dimensionless (meters/meters) One complete rotation equals 2 radiansOne radian equals 57.3 degrees
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Angular DisplacementAngular Displacement
Angular displacement (), angle of rotation measured in radians
Linear displacement equals angular displacement times the radiusAll parts of rotating object have same angular displacement
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Angular VelocityAngular Velocity
Change in angular displacement with respect to time = t; fundamental units are rad/s, but usually measured in rev/s, or rev/min (rpm)Vector with direction found using right hand rule: Curl fingers of right hand in direction of rotation. Extended thumb points in direction of vector
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Angular VelocityLinear velocity = angular velocity times radius, v = r
All parts of rotating object have same angular velocity
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Angular AccelerationAngular Acceleration
Change in angular velocity with respect to time: t; units are rad/s2
Vector whose direction is found using right hand ruleIf angular acceleration is constant, constant acceleration equations are used.
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Comparing Linear and Angular VariablesComparing Linear and Angular Variables
Quantity Linear Angular Conversion Displacement d (x, y) d = r
Velocity v v = r Acceleration a a = r
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Constant Acceleration EquationsConstant Acceleration Equations
Linearvf = vi + at
d = vit + 1/2(at2)
vf2 = vi
2 + 2ad
Angularf = i + t
it + 1/2(t2)f
2 = i2 + 2
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Center of GravityThe point at which all object’s weight can be considered to be concentrated.For symmetrical bodies with uniform density, c.o.g. will be at geometric center. May be located outside the body of some objects.Bodies or systems rotate about their center of gravity.Similar to center of mass but not always the same
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Parallel Forces
Forces acting in the same or opposite directions at different points on an objectCan produce rotationConcurrent forces act at the same point (often the center of gravity) at the same time on an object
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Weight Vectors
Drawn from center of gravity of objectActually are the sum of an infinite number of parallel weight vectors from an infinite number of mass unitsThe effect is as if all the weight was concentrated at the center of gravity
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TorqueThe result of a force that produces rotation, a vector The product of the force and its lever arm,Lever arm (or moment arm) is a vector whose magnitude is the distance from the point of rotation to the point of application of the force= r x F A product of two vectors that produces a third perpendicular vector
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Torque units are meters x newtons signs: ccw torques are considered +, cw torques are -direction of net torque is direction of resulting rotation
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Rotational InertiaRotational Inertia
Resistance of an object to any change in angular velocityDepends on mass and its orientation with respect to axis of rotation.Is rotational analogue to mass; symbol I, units kg m2
Sometimes called moment of inertia
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Rotational InertiaFor an object rotating about an external point, I = mr2
For objects rotating about an internal axis, inertia must be calculated using calculusUse rotational inertia equations for general type of regularly shaped solid bodies
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Newton’s Second Law for RotationNewton’s Second Law for Rotation
Substitute angular variables for linearF = ma becomes = Iwhere is the net torque and I is the rotational inertia of the body.
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Work in Rotary MotionWork in Rotary Motion
Work done by torqueW = = Fr is angular displacement in radiansAssumes force is perpendicular to radius
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Power in Rotary MotionPower in Rotary Motion
Power is rate of doing workP = /t
/t = , so P =
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Kinetic Energy in Rotary MotionKinetic Energy in Rotary Motion
Energy possessed by rotating objectKErot = 1/2(I2)
Rolling objects have both linear and rotational kinetic energy
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Kinetic Energy in Rotary MotionWhen object rolls downhill, potential energy is converted to both types of kinetic energy; amount of each depends on rotational inertia of object.
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Angular Momentum
The tendency of a rotating object to continue rotatingA combination of the rotational inertia and angular velocityFor a rotating object, L = IA vector
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Angular Momentum
Objects in circular motion also have angular momentum: L = mvr
Angular momentum can be applied to any moving object with respect to an external pointRadial distance is perpendicular distance form path of object to the point
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Conservation of Angular MomentumExternal net torque is required to change angular momentumIf no net external torque is present, angular momentum of a system will remain constantTotal angular momentum before the interaction equals total angular momentum after the interaction as long as no net external torque acts on the system
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Conservation of Angular MomentumAlways true, from atomic to galactic interactionsIf rotational inertia changes, angular velocity must change to conserve angular momentum
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PrecessionPrecession
A secondary rotation of the axis of rotationDue to torque produced by weight of rotating objectCauses angular acceleration that changes direction of angular velocity of rotating objectEarth precesses on its axis with a secondary rotation period of 26,000 years