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Unformatted text preview: Chapter 8
Rotational Motion Angular Quantities Radian angle whose arc length is equal to the radius. More useful than degrees when describing rotational motion. = lr 1 revolution = 360 degrees = 2 radians Convert 45 degrees to radians. 2.5 radians to degrees 3 revolutions to degrees and radians. Examples: Angular Quantities Angular displacement () change in angular position for a point on a rotating object. Analogous to displacement (x). = 2 - 1 Angular velocity () angular displacement over a certain time interval. Unit: radians Angular acceleration () change in angular velocity over a certain time interval. Analogous to velocity (v). Unit: rad/sec = t Analogous to acceleration (a). Unit: rad/sec2 = t Angular Quantities Relationships to linear quantities: Displacement: Velocity: x = r Acceleration: v = r
atan = r ac = r
2 All points on a rigid rotating object experience the same angular displacement, velocity, and acceleration but may experience different linear displacement, velocity, and acceleration. Angular Quantities Frequency number of complete rotations in one second. Unit: 1 Hertz= 1 rev/sec = 2f
T= 1 f Period time for one complete rotation. Unit: seconds Rotational Kinematics with Constant The equations to describe the motion of a rigid object rotating at a constant velocity are very similar to the equations we used to describe linear motion. = 0 + t
2 = 0t + 2 t
2 = - 2 2 0 Rotational Kinematics with Constant Example: A wheel moving at 2.0 rad/s undergoes a constant angular acceleration of 3.5 rad/s2. Find the angle it has rotated through and its angular velocity after 5.0 seconds. How far will a point 3 cm from the wheel's center have moved during this time? Example: A car engine revs up from 0 rpm to 1200 rpm in 2.5 seconds. What is its angular acceleration and how many times does it rotate? Torque Torque () tendency of a force to cause rotation. Depends on the distance of the force from the pivot point. This distance is called the lever arm. Depends on the component of the force that is perpendicular to the lever arm. = rF sin = rF Unit of Torque: m N Note: this is NOT the same as N m = Joule Torque Net Torque () sum of all torques acting on an object. A net torque causes an angular acceleration. Torque is the angular analogue to force. Torque has a sign determined by which direction it would tend to cause an object to rotate. Typically, counterclockwise torques are positive and clockwise torques are negative. Torque Example: Calculate the net torque on the meter stick below if FA is 25 N and FB is 15 N. Torque Example: Two disks with different radii (rA = 30 cm and rB=50 cm) are attached together as shown. Calculate the net torque and determine the direction of rotation for the two disks. Rotational Dynamics Moment of Inertia (I) rotational equivalent of mass. Resistance to a change in rotation. For a single mass m rotating in a circle with radius r : I=mr2 For an extended object, the moment of inertia must be found by summing the individual moments of all parts of the object using calculus. See pg 208 for the moments for some common shapes. Depends on the mass and the distribution of mass. Rotational Dynamics Newton's Second Law for Rotation: = I Question: A ring and a disk of equal mass and radius are both released from rest at the top of an inclined plane. Which one reaches the bottom first?\ Example: A circular saw has a blade that is 25 cm in diameter with a mass of 0.75 kg. How much torque is needed to bring it from rest to 500 rpm in 0.5 seconds? Angular Momentum Angular Momentum (L) Similar to linear momentum, measure of how difficult it is to stop a rotating object. Depends on moment of inertia and angular velocity. L = I Conservation of Angular Momentum the total angular momentum of a rotating object remains constant if the net torque acting on it is zero. Angular Momentum Direction of Angular Vectors Examples: By convention, the direction of the angular velocity vector is found by the "righthand rule": If the fingers of your right hand curl in the direction of rotation, the thumb points in the direction of the angular velocity. Skater pulling in arms during a spin Star collapse Stability of a spinning top, Gyroscope ...
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This note was uploaded on 04/17/2008 for the course PH 1113 taught by Professor Winter during the Fall '07 term at Mississippi State.
- Fall '07