Set 4 Circles and NewtonWhere Are WeRemember from the past …Changing VelocityUniform Circular MotionQuick Review - RadiansChanging Velocity in Uniform Circular MotionThe accelerationCentripetal AccelerationCentripetal Acceleration, contPeriodTangential AccelerationTotal AccelerationTotal Acceleration, equationsTotal Acceleration, In Terms of Unit VectorsSlide 16Slide 17Set 4 Set 4 Circles and NewtonCircles and NewtonFebruary 3, 2006February 3, 2006Where Are WeWhere Are We•Today–Quick review of the examination– we finish one topic from the last chapter – circular motion•We then move on to Newton’s Laws•New WebAssign on board on today’s lecture material–Assignment – Read the circular motion stuff and begin reading Newton’s Laws of Motion•Next week–Continue Newton–Quiz on Friday•Remember our deal!Remember from the past …Remember from the past …•Velocity is a vector with magnitude and direction.•We can change the velocity in three ways–increase the magnitude–change the direction–or both•If any of the components of v change then there is an acceleration.Changing VelocityChanging Velocityv1v2v2vaUniform Circular MotionUniform Circular Motion•Uniform circular motion occurs when an object moves in a circular path with a constant speed•An acceleration exists since the direction of the motion is changing –This change in velocity is related to an acceleration•The velocity vector is always tangent to the path of the objectQuick Review - RadiansQuick Review - RadianssRadians rsChanging Velocity in Changing Velocity in Uniform Circular MotionUniform Circular Motion•The change in the velocity vector is due to the change in direction•The vector diagram shows v = vf - viThe accelerationThe acceleration 2rvarvttvratvtvvvCentripetalAccelerationCentripetal AccelerationCentripetal Acceleration•The acceleration is always perpendicular to the path of the motion•The acceleration always points toward the center of the circle of motion•This acceleration is called the centripetal accelerationCentripetal Acceleration, Centripetal Acceleration, contcont•The magnitude of the centripetal acceleration vector was shown to be•The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion2Cvar=PeriodPeriod•The period, T, is the time required for one complete revolution•The speed of the particle would be the circumference of the circle of motion divided by the period•Therefore, the period is 2 rTvp�Tangential AccelerationTangential Acceleration•The magnitude of the velocity could also be changing•In this case, there would be a tangential accelerationTotal AccelerationTotal Acceleration•The tangential acceleration causes the change in the speed of the particle•The radial acceleration comes from a change in the direction of the velocity vectorTotal Acceleration, Total Acceleration, equationsequations•The tangential acceleration:•The radial acceleration:•The total acceleration:–Magnitude tdadt=v2r Cva ar=- =-2 2r ta a a= +Total Acceleration, In Terms Total Acceleration, In Terms of Unit Vectorsof Unit Vectors•Define the following unit vectors–r lies along the radius vector is tangent to the circle•The total acceleration isˆˆandr q2ˆˆt rdvdt rq= + = -va a a rA ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m. The plane of the circle is 1.20 m above the ground. The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the ball's location when the string breaks. Find the radial acceleration of the ball during its circular motion.122rrvA pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.53). When the pendulum is in the two horizontal positions = 90.0° and = 270°, its speed is 5.00 m/s. (a) Find the magnitude of the radial acceleration and
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