Ying Yi PhD Chapter 5 Dynamics of Uniform Circular Motion 1 PHYS HCC

Outline 2 What is Uniform Circular Motion Centripetal Acceleration Centripetal Force PHYS HCC

Circular Motion PHYS HCC 3

Uniform Circular Motion PHYS HCC 4 Speed doesn’t change!

PHYS HCC 5 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 6 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 7 PHYS HCC

Is this a rigid body? PHYS HCC 8

9 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 10 PHYS HCC

11 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 12 PHYS HCC

Example: Angular velocity PHYS HCC 13 The rotor on a helicopter turns at an angular speed of 3.20×10 2 revolutions per minute (rpm). (a) Find the period of this rotation (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?

Group Problem: Angular velocity PHYS HCC 14 The wheel of a car has a radius of r=0.29 m and is being rotated at 830 revolution per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving.

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 15 PHYS HCC

16 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 17 PHYS HCC

Example 5.3 Bobsled Track PHYS HCC 18 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.

Forces Causing Centripetal Acceleration  Tension in a string  Force of friction  Gravitational force 19 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 20 PHYS HCC

21 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 5.5 Tension PHYS HCC 22 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 23 When the car moves without skidding around a curve, static friction between the road and the tires provides the centripetal force to keep the car on the road. Centripetal Force Example 2: Friction

Example 5.7 Safe Driving PHYS HCC 24 At what maximum speed can a car safely negotiate a horizontal unbanked turn (radius 51 m) in Dry weather (coefficient of static friction =0.95)?

Group Problem: Friction PHYS HCC 25 At what maximum speed can a car safely negotiate a horizontal unbanked turn (radius 51 m) in icy weather (coefficient of static friction =0.10)?

PHYS HCC 26 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.

Example 5.8 Banked curve PHYS HCC 27 The Daytona 500 is the major event of the NASCAR season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius of r=316 m and are banked steeply, with Ɵ =31º. Suppose these maximum-radius turns were frictionless. At what speed would the cars have to travel around them?

PHYS HCC 28 Centripetal Force Example 4: Gravitational force For a satellite in circular orbit around the earth, the gravitational force provides the centripetal force.

Group Problem: Hubble Telescope PHYS HCC 29 Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km aboe the earth’s surface. (M E =5.98×10 24 kg, G=6.67× Nm 2 /kg 2 )

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 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) 30 PHYS HCC

Homework PHYS HCC 31 2,9,12,13,16,23,28,31