7 1 Rotational Motion and Law of Gravity Smooth circular objects Kinematics in circles Will he catch the turtle x x A Linear motion He can reach out 1m x displacement x is the same for everyone B Rotational motion angle is the same for everyone while linear distances and speeds Reference line are not Page 1 Will he catch theposition turtle Angle angular He can reach out 1m A Definition r s arc length r radius s Reference line unit rad Counter clockwise Clockwise decreasing Circumference r 2 Conversion rad to degree 1 revolution 1 rad 360 2 s 2 r 2 rad 360 r r Example The Sun s r 0 5 The Moon s r 0 5 Total solar eclipse 2 160 mi 864 000 mi 240 000 mi 93 000 000 mi Page 2 Angular Displacement D 0 Angular Velocity Average angular velocity angular displacement elapsed time wavg D Dt 0 t t0 Angular Acceleration Average angular acceleration change in velocity elapsed time aavg wf w0 t t0 Dw Dt Page 3 Example A floating volleyball serve To make a volleyball serve float the server has to spin the ball for a fraction of a second a good example of aerial dynamics The ball goes from 1 0 rad s to 2 1 0 rad s during t 0 1 s Find a 7 2 The Equation of Rotational Motion with comparison to 1 D linear kinematics Angular Linear a constant a constant v v 0 at 0 t 0 w 0 t 1 2 at 2 w 2 w 2 2a 0 x x 0 v 0t v2 v02 2 ax Page 4 1 2 at 2 Example 0 36 0 rad s 34 2 rad s t 0 595 s a 3 03 rad s2 0 t How many revolutions does the ball make before being caught 0 0 w 0 t 1 2 at 2 0 36 0 0 595 0 5 3 03 0 5952 20 88 rad 1 rev 2 rad or 1 rad 1 2 rev 20 88 2 3 3 rev Quiz Qn 2 A 0 3m radius wheel rotates how many radians after starting from rest and accelerates at a constant acceleration of 2 rad s2 over a 5 s interval Page 5 7 3 Angular and Linear Quantities s r v Ds D r Dt Dt v rw at Dv Dw r Dt Dt at ra s a r v r r x s Quiz Qn 3 A grindstone spinning at the rate of 8 3 rev s has what approximate angular speed Page 6 7 4 Centripetal acceleration There is always a centripetal acceleration caused by changing of direction in rotational motion rw 2 r 2w 2 v2 ac r r r ac rw 2 at Tangential Acceleration at r Total Acceleration ac a at2 ac2 at r Tangential Acceleration at a Total Acceleration ac a at2 ac2 at r can be both positive and negative The sign doesn t affect the magnitude of a but changes the direction Page 7 Example Discus thrower turns with angular acceleration t 50 rad s2 Find the total acceleration when the angular velocity is 10 rad s ac at r 0 8 50 40 m s2 ac r 2 0 8 102 80 m s2 at total acceleration magnitude a at2 ac2 89m s 2 Vector Nature of Angular Quantities Angular displacement velocity and acceleration are all vector quantities Direction can be more completely defined by using the right hand rule Grasp the axis of rotation with your right hand Wrap your fingers in the direction of rotation Your thumb points in the direction of Page 8 Velocity Directions In a the disk rotates clockwise the velocity is into the page In b the disk rotates counterclockwise the velocity is out of the page Forces Causing Centripetal Acceleration Newton s Second Law says that the centripetal acceleration is accompanied by a force FC maC FC stands for any force that keeps an object following a circular path Tension in a string Gravity Force of friction Page 9 Quiz Qn 4 What centripetal force does an 80 kg passenger experience when seated 12 m from the center of a Ferris wheel whose angular speed is 0 5 rad s Level Curves Friction is the force that produces the centripetal acceleration Can find the frictional force or v v rg Page 10 Banked Curves A component of the normal force adds to the frictional force to allow higher speeds v2 tan rg or ac g tan Vertical Circle Look at the forces at the top of the circle The minimum speed at the top of the circle can be found v top gR Page 11 Example 1 A car travels at a constant speed of 30 miles hr on a level circular turn of radius 50 m What is the minimum coefficient of friction between the tires and the road will allow the car to make this turn without sliding Quiz Qn 5 At what angle relative to the horizontal should a curve 52 m in radius be banked if no friction is required to prevent the car from slipping when traveling at 12 m s Page 12 Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them F G m1m2 r2 G 6 673 x 10 11 N m kg Page 13