Ying Yi PhD Chapter 7 Rotational Motion and the Law of Gravity 1 PHYS HCC

Circular Motion PHYS HCC 2

Outline 3 Angular Speed and Angular Acceleration Constant Angular Acceleration Motion Relation between Angular and Linear Quantities Centripetal Acceleration PHYS HCC

4 The Radian The radian is a unit of angular measure The radian can be defined as the arc length along a circle divided by the radius r

More About Radians Comparing degrees and radians Converting from degrees to radians 5 PHYS HCC

Rigid Body Every point on the object undergoes circular motion about the point O All parts of the object of the body rotate through the same angle during the same time The object is considered to be a rigid body This means that each part of the body is fixed in position relative to all other parts of the body 6 PHYS HCC

Is this a rigid body? PHYS HCC 7

8 Angular Displacement Axis of rotation is the center of the disk Need a fixed reference line During time t, the reference line moves through angle θ

Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement 9 PHYS HCC

10 Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Angular Speed, cont. The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero Units of angular speed are radians/sec rad/s Speed will be positive if θ is increasing (counterclockwise) Speed will be negative if θ is decreasing (clockwise) 11 PHYS HCC

Example 7.1 Whirlybirds PHYS HCC 12 The rotor on a helicopter turns at an angular speed of 3.20×10 2 revolutions per minute (rpm). (a) Express this angular speed in radians per second. (b) If the rotor has a radius of 2.00 m, what arclength does the tip of the blade trace out in 3.00×10 2 s?

Average Angular Acceleration The average angular acceleration 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: 13 PHYS HCC

Angular Acceleration, cont Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration 14 PHYS HCC

Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular speed The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero 15 PHYS HCC

16 Linear and Rotational Motion

Example 7.2 A rotating Wheel PHYS HCC 17 A wheel rotates with a constant angular acceleration of 3.50 rad/s 2. If the angular speed of the wheel is 2.00 rad/s at t=0, (a) through what angle does the wheel rotate between t=0 and t=2.00s? Give your answer in radians and in revolutions. (b) What is the angular speed of the wheel at t=2.00 s? (c) What angular displacement (in revolutions) results while the angular speed found in part (b) doubles?

Relationship Between Angular and Linear Quantities Displacements Speeds Accelerations Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion 18 PHYS HCC

Example 7.4 Track Length of a compact disc PHYS HCC 19 In a compact disc player, as the read head moves out from the center of the disc, the angular speed of the disc changes so that the linear speed at the position of the head remains at a constant value of about 1.3 m/s (a) find the angular speed of the compact disc when the read head is at r=2.0 cm and again at r=5.6 cm. (b) An old-fashioned record player rotates at a constant angular speed, so the linear speed of the record groove moving under the detector changes. Find the linear speed of a 45.0 rpm record at points 2.0 and 5.6 cm from the center. (c) In both the CD and phonograph record, information is recorded in a continuous spiral track, 1.3 m/s. Calculate the total length of the track for a CD designed to play for 1.0 h.

Group Problem: Compact Discs PHYS HCC 20 A compact disc rotates from rest up to an angular speed of 31.4 rad/s in a time of s. (a) What is the angular acceleration of the disc, assuming the angular acceleration is uniform? (b) Through what angle does the disc turn while coming up to this speed? (c) If the radius of the disc is 4.45 cm, find the tangential speed of a microbe riding on the rim of the disc when t=0.892 s. (d) What is the magnitude of the tangential acceleration of the microbe at the given time?

Uniform Circular Motion PHYS HCC 21 Speed doesn’t change!

Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration is due to the change in the direction of the velocity 22 PHYS HCC

23 Centripetal Acceleration, cont. Centripetal refers to “center-seeking” The direction of the velocity changes The acceleration is directed toward the center of the circle of motion

Centripetal Acceleration, final The magnitude of the centripetal acceleration is given by This direction is toward the center of the circle 24 PHYS HCC

Centripetal Acceleration and Angular Velocity The angular velocity and the linear velocity are related (v = ω r) The centripetal acceleration can also be related to the angular velocity 25 PHYS HCC

Group Problem: Bobsled Track PHYS HCC 26 The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as Figure 5.5 illustrates. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. Express the answers as multiples of g=9.8 m/s 2.

Total Acceleration The tangential component of the acceleration is due to changing speed The centripetal component of the acceleration is due to changing direction Total acceleration can be found from these components 27 PHYS HCC

28 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 ω

PHYS HCC 29 Velocity Directions, Example 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

Acceleration Directions If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with time If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time 30 PHYS HCC

Example 7.5 At the Racetrack PHYS HCC 31 A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s while traveling counterclockwise around a circular track of radius 4.00×10 2 m. When the car reaches a speed of 50.0 m/s, find (a) the magnitude of the car’s centripetal acceleration, (b) the angular speed, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration.

Forces Causing Centripetal Acceleration  Tension in a string  Force of friction  Gravitational force 32 PHYS HCC

Centripetal Force General equation If the force vanishes, the object will move in a straight line tangent to the circle of motion Centripetal force is a classification that includes forces acting toward a central point It is not a force in itself 33 PHYS HCC

34 Centripetal Force Example 1: Tension A ball of mass m is attached to a string Its weight is supported by a frictionless table The tension in the string causes the ball to move in a circle

Example: Tension PHYS HCC 35 The model airplane in figure 5.6 has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. The path of the airplane and its guideline lie in the same horizontal plane, because the weight of the plane is balanced by the lift generated by its wings. Find the tension in the guideline (length=17 m) for speeds of 19 and 38 m/s.

PHYS HCC 36 Friction is the force that produces the centripetal acceleration Can find the frictional force, µ, or v Centripetal Force Example 1:Friction

Example 7.6 Buckle Up for Safety PHYS HCC 37 A car travels at a constant speed of 30.0 mi/h (13.4 m/s) on a level circular turn of radius 50.0 m. What minimum coefficient of static friction, µ s, between the tires and roadway will allow the car to make the circular turn without sliding?

PHYS HCC 38 Centripetal Force Example 3: Normal force A car travels on a circle of radius r on a frictionless banked road. The banking angle is Ɵ, and the center of the circle is at C. The force acting on the car are its weight and normal force. A component F N sin Ɵ of the normal force provide the centripetal force.

Group Problem: Banked curves PHYS HCC 39 The Daytona international Speedway in Daytona Beach, Florida, is famous for its races especially the Daytona 500, held every February. Both of its courses feature four-story, 31.0º banked curves, with maximum radius of 316 m. If a car negotiates the curve too slowly, it tends to slip down the incline of the turn, whereas if it’s going too fast, it may begin to slide up the incline. (a) Find the necessary centripetal acceleration on this banked curve so the car won’t slip down or slide up the incline. (Neglect friction) (b) Calculate the speed of the car.

Problem Solving Strategy Draw a free body diagram, showing and labeling all the forces acting on the object(s) Choose a coordinate system that has one axis perpendicular to the circular path and the other axis tangent to the circular path The normal to the plane of motion is also often needed 40 PHYS HCC

Problem Solving Strategy, cont. Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, F C ) Use Newton’s second law The directions will be radial, normal, and tangential The acceleration in the radial direction will be the centripetal acceleration Solve for the unknown(s) 41 PHYS HCC

Thank you PHYS HCC 42