Circular Motion

Angle and Radian What is the circumference S ? q can be defined as the arc length s along a circle divided by the radius r: q is a pure number, but commonly is given the artificial unit, radian (“rad”) r s r s   Whenever using rotational equations, you must use angles expressed in radians

Conversions Comparing degrees and radians Converting from degrees to radians Converting from radians to degrees January 20, 2009

Rigid Object A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible This simplification allows analysis of the motion of an extended object

One Dimensional Position x What is motion? Change of position over time. How can we represent position along a straight line? Position definition: Defines a starting point: origin (x = 0), x relative to origin Direction: positive (right or up), negative (left or down) It depends on time: t = 0 (start clock), x(t=0) does not have to be zero. Position has units of [Length]: meters. x = + 2.5 m x = - 3 m

Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin As the particle moves, the only coordinate that changes is q As the particle moves through q, it moves though an arc length s. The angle q, measured in radians, is called the angular position. January 20, 2009

Displacement Displacement is a change of position in time. f stands for final and i stands for initial. It is a vector quantity. It has both magnitude and direction: + or - sign It has units of [length]: meters. x1 (t1) = + 2.5 m x2 (t2) = - 2.0 m Δx = -2.0 m - 2.5 m = -4.5 m x1 (t1) = - 3.0 m x2 (t2) = + 1.0 m Δx = +1.0 m + 3.0 m = +4.0 m

Angular Displacement The angular displacement is defined as the angle the object rotates through during some time interval SI unit: radian (rad) This is the angle that the reference line of length r sweeps out

Ch5-Circular Motion-Revised 2/15/10 Arc Length x y i f  r arc length = s = r  is a ratio of two lengths; it is a dimensionless ratio! MFMcGraw Ch5-Circular Motion-Revised 2/15/10

While riding a carousel that is rotating clockwise, a child travels through and arc length of 11.5 m. If the child’s angular displacement is 165o, what is the radius of the carousel? Given: Δθ=- 165o Δs=-11.5m Δr=? Δθ(rad)=(π/180) Δθ(deg)=-2.88 rad Δθ = Δs/r  r=Δs/ Δθ =-11.5 m/-2.88 rad = 3.99 m

Velocity Velocity is the rate of change of position. Velocity is a vector quantity. Velocity has both magnitude and direction. Velocity has a unit of [length/time]: meter/second. Definition: Average velocity Average speed Instantaneous velocity

Average and Instantaneous Angular Speed The average angular speed, ωavg, of a rotating rigid object is the ratio of the angular displacement to the time interval The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero SI unit: radian per second (rad/s) Angular speed positive if rotating in counterclockwise Angular speed will be negative if rotating in clockwise

Ch5-Circular Motion-Revised 2/15/10 Angular Speed x y i f r  An object moves along a circular path of radius r; what is its average speed? Also, (instantaneous values). MFMcGraw Ch5-Circular Motion-Revised 2/15/10

A child at an ice cream parlor spins on a stool A child at an ice cream parlor spins on a stool. The child turns counter clockwise with an average angular speed of 4 rad/s. In what time interval will the child’s feet have an angular displacement of 8 π rad? Given: Δθ= 8 π rad ωavg = 4 rad/s Δt=? ωavg = Δθ /Δt  Δt= Δθ / ωavg = 8 π rad/ 4 rad/s=6.3 s

Average Acceleration Changing velocity (non-uniform) means an acceleration is present. Acceleration is the rate of change of velocity. Acceleration is a vector quantity. Acceleration has both magnitude and direction. Acceleration has a unit of [length/time2]: m/s2. Definition: Average acceleration Instantaneous acceleration

Average Angular Acceleration The average angular acceleration, a, of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: wi wf

A car’s tire rotates at an initial angular speed of 21. 5 rad/s A car’s tire rotates at an initial angular speed of 21.5 rad/s. The driver accelerates, and after 3.5 s the tire’s angular speed is 28 rad/s. What is the tire’s average angular acceleration? Given: ω1= 21.5 rad/s ω2=28 rad/s Δt=3.5 s αavg =? αavg = ω2 – ω1/t2 – t1 = Δω/Δt αavg = 28 rad/s – 21.5 rad/s /3.5 s= 1.9 rad/s2

Period and Frequency The time it takes to go one time around a closed path is called the period (T). Comparing to v = r: f is called the frequency, the number of revolutions (or cycles) per second.

Centripetal Acceleration Consider an object moving in a circular path of radius r at constant speed. x y v Here, v  0. The direction of v is changing. If v  0, then a  0. Then there is a net force acting on the object. Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 5, Questions 4 and 7.

Centripetal Acceleration Conclusion: with no net force acting on the object it would travel in a straight line at constant speed It is still true that F = ma. But what acceleration do we use?

Centripetal Acceleration The velocity of a particle is tangent to its path. For an object moving in uniform circular motion, the acceleration is radially inward.

Centripetal Acceleration The magnitude of the radial acceleration is:

Circular Motion and N.S.L Recall that according to Newton’s Second Law, the acceleration is directly proportional to the Force. If this is true: Since the acceleration and the force are directly related, the force must ALSO point towards the center. This is called CENTRIPETAL FORCE. NOTE: The centripetal force is a NET FORCE. It could be represented by one or more forces. So NEVER draw it in an F.B.D.

Examples The blade of a windshield wiper moves through an angle of 90 degrees in 0.28 seconds. The tip of the blade moves on the arc of a circle that has a radius of 0.76m. What is the magnitude of the centripetal acceleration of the tip of the blade?

Examples What is the minimum coefficient of static friction necessary to allow a penny to rotate along a 33 1/3 rpm record (diameter= 0.300 m), when the penny is placed at the outer edge of the record? Top view FN Ff mg Side view

Examples The maximum tension that a 0.50 m string can tolerate is 14 N. A 0.25-kg ball attached to this string is being whirled in a vertical circle. What is the maximum speed the ball can have (a) the top of the circle, (b)at the bottom of the circle? T mg

Examples At the bottom? T mg

Ch5-Circular Motion-Revised 2/15/10 Unbanked Curve A coin is placed on a record that is rotating at 33.3 rpm. If s = 0.1, how far from the center of the record can the coin be placed without having it slip off? We’re looking for r. x y w N fs Draw an FBD for the coin: Apply Newton’s 2nd Law: MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Ch5-Circular Motion-Revised 2/15/10 Unbanked Curve From (2) Solving for r: What is ? MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Ch5-Circular Motion-Revised 2/15/10 Banked Curves A highway curve has a radius of 825 m. At what angle should the road be banked so that a car traveling at 26.8 m/s has no tendency to skid sideways on the road? (Hint: No tendency to skid means the frictional force is zero.)  Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 5, Questions 8, 18, and 19. Take the car’s motion to be into the page. MFMcGraw Ch5-Circular Motion-Revised 2/15/10

On a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight.

Ch5-Circular Motion-Revised 2/15/10 Banked Curves x y N w  FBD for the car: Apply Newton’s Second Law: MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Ch5-Circular Motion-Revised 2/15/10 Banked Curves Rewrite (1) and (2): Divide (1) by (2): MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Example The turns at the Daytona International Speedway have a maximum radius of 316 m and are steely banked at 31 degrees. Suppose these turns were frictionless. As what speed would the cars have to travel around them?

Ch5-Circular Motion-Revised 2/15/10 Circular Orbits Consider an object of mass m in a circular orbit about the Earth. Earth r The only force on the satellite is the force of gravity: Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 5, Questions 9 and 20. Solve for the speed of the satellite: MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Ch5-Circular Motion-Revised 2/15/10 Circular Orbits Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours? From previous slide: Also need, Combine these expressions and solve for r: MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Ch5-Circular Motion-Revised 2/15/10 Circular Orbits Kepler’s Third Law It can be generalized to: Where M is the mass of the central body. For example, it would be Msun if speaking of the planets in the solar system. MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Rotational Kinematics

Centripetal vs. Centrifugal The centrifugal force is not a “real” force — the tendency to fly outwards is observed because objects that are moving in a straight line tend to continue moving in a straight line. This is called inertia, and it makes objects resistant to the force that makes them move in a curve.

Nonuniform Circular Motion Nonuniform means the speed (magnitude of velocity) is changing. at v ar a There is now an acceleration tangent to the path of the particle. Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 5, Questions 5, 10, 11, 12, and 15. The net acceleration of the body is This is true but useless! MFMcGraw Ch5-Circular Motion-Revised 2/15/10

Nonuniform Circular Motion at ar a at changes the magnitude of v. Changes energy - does work ar changes the direction of v. Doesn’t change energy - does NO WORK The accelerations are only useful when separated into perpendicualr and parallel components. Can write: MFMcGraw Ch5-Circular Motion-Revised 2/15/10

A Brief Review from Chapter 5 Angular displacement: Units: radians (rad)

A Brief Review from Chapter 5 Average angular velocity: units: rad/s or: degrees/s, rev/min, etc.

Angular Acceleration Average angular acceleration: units: rad/s2 or: degrees/s2, rev/min2, etc.

Rotational Kinematic Equations Definition of average angular velocity:

Rotational Kinematic Equations Definition of average angular acceleration:

Rotational Kinematic Equations A previous result:

Rotational Kinematic Equations Solve definition of average acceleration for t: Substitute into a previous result:

Comparison: Kinematic Equations Rotational Linear (a = constant) (a = constant)

Comparison: Kinematic Equations Same equations, (some) different variables Position, displacement: x q Time: t t Velocity, speed: v w Acceleration: a a

Angular and Tangential Velocity Average angular velocity is the angular displacement divided by the time interval in which it occurred.

Angular and Tangential Acceleration From the definition of linear acceleration: From the definition of angular acceleration: Combining:

Angular Velocity, Centripetal Acceleration From chapter 5: But: Substituting:

Total Acceleration The tangential and centripetal accelerations are vector components of the total acceleration.

Angular Vectors Angular displacement, q, is not a vector quantity. the reason: addition of angular displacements is not commutative. Where you end up depends on the order in which the angular displacements (rotations) occur.

Angular Vectors Angular velocity, w, and angular acceleration, a, are vectors. Magnitudes: and Directions: Parallel to the axis of rotation, and in the direction given by the right-hand rule:

Angular Vectors Right-hand rule direction for w:

Angular Vectors Right-hand rule direction for a: Also parallel to axis of rotation Same direction as change in w vector Same direction as w if w is increasing in magnitude Opposite direction from w if w is decreasing in magnitude