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Chapter 4 Lecture Week 12 Day 1 Circular Motion © 2014 Pearson Education, Inc. Circular Motion • Why do pilots sometimes black out while pulling out at the bottom of a power dive? • Are astronauts really "weightless" while in orbit? • Why do you tend to slide across the car seat when the car makes a sharp turn? © 2014 Pearson Education, Inc. Centrifugal Force • If the car you are in turns a corner quickly, you feel “thrown” against the door. • The “force” that seems to push an object to the outside of a circle is called the centrifugal force. • This is caused by your inertia pressing you against the door Fictitious Forces • If you are riding in a car that makes a sudden stop, you may feel as if a force “throws” you forward toward the windshield. • There really is no such force. • Nonetheless, the fact that you seem to be hurled forward relative to the car is a very real experience! • You can describe your experience in terms of what are called fictitious forces or pseudo forces. . Reading Quiz 1. For uniform circular motion, the acceleration A. B. C. D. E. is parallel to the velocity. is directed toward the center of the circle. is larger for a larger orbit at the same speed. is always due to gravity. is always negative. Slide 6-6 Answer 1. For uniform circular motion, the acceleration A. B. C. D. E. is parallel to the velocity. is directed toward the center of the circle. is larger for a larger orbit at the same speed. is always due to gravity. is always negative. Slide 6-7 Reading Quiz 2. When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? A. B. C. D. E. Friction Tension Normal force Air resistance Gravity Slide 6-8 Answer 2. When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? A. B. C. D. E. Friction Tension Normal force Air resistance Gravity Slide 6-9 Radial acceleration: An object at constant speed v on a circular path of radius r • Based on our testing experiments, we found: • The constant of proportionality is 1, so we can define the radial acceleration as: • The acceleration points toward the center. • The SI units are m/s2. • In the limiting case of a straight line, the radius goes to infinity and the acceleration goes to zero. This equation makes sense. © 2014 Pearson Education, Inc. Every quantity we studied has a rotational analog We have mainly discussed particle motion • This describes motion of the center of mass • What about rotational motion about the center of mass • Every motion quantity we have looked at in kinematics, Forces, momentum, and Energy has an analogous quantity in Rotation. • Every equation has an analog too • Rotational equations look exactly the same • Examples • Position => theta • Velocity => angular velocity • Force => torque • Newton’s 2nd Law => Newton’s 2nd Law for rotational motion • Mass => Moment of Inertia Slide 15-37 Units of rotational position • The unit for rotational position is the radian (rad). It is defined in terms of: – The arc length s – The radius r of the circle • The angle in units of radians is the ratio of s and r: • The radian unit has no dimensions; it is the ratio of two lengths. The unit rad is just a reminder that we are using radians for angles. © 2014 Pearson Education, Inc. Tip © 2014 Pearson Education, Inc. Rotational (angular) velocity ω • Translational velocity is the rate of change of linear position. • We define the rotational (angular) velocity v of a rigid body as the rate of change of each point's rotational position. – All points on the rigid body rotate through the same angle in the same time, so each point has the same rotational velocity. © 2014 Pearson Education, Inc. Rotational (angular) velocity ω © 2014 Pearson Education, Inc. Tips © 2014 Pearson Education, Inc. Rotational (angular) acceleration α • Translational acceleration describes an object's change in velocity for linear motion. – We could apply the same idea to the center of mass of a rigid body that is moving as a whole from one position to another. • The rate of change of the rigid body's rotational velocity is its rotational acceleration. – When the rotation rate of a rigid body increases or decreases, it has a nonzero rotational acceleration. © 2014 Pearson Education, Inc. Rotational (angular) acceleration α © 2014 Pearson Education, Inc. Tip © 2014 Pearson Education, Inc. Torque τ produced by a force • The SI unit of force is the Newton-meter (N-m). © 2014 Pearson Education, Inc. Tip © 2014 Pearson Education, Inc. Tip • To decide the sign of the torque, pretend that a pencil is the rigid body. Hold it with two fingers at a place that represents the axis of rotation, and exert a force on the pencil representing the force whose torque sign you wish to determine. Does the force cause the pencil to turn counterclockwise (a positive torque) or clockwise (a negative torque) about the axis of rotation? © 2014 Pearson Education, Inc. Conditions of equilibrium • An object remains at rest with respect to a particular observer in an inertial reference frame. • In this chapter, we consider only observers who are at rest with respect to Earth. © 2014 Pearson Education, Inc. Condition 2: Rotational (torque) condition of equilibrium • A rigid body is in turning or rotational static equilibrium if it is at rest with respect to the observer and the sum of the torques (positive counterclockwise torques and negative clockwise torques) about any axis of rotation produced by the forces exerted on the object is zero. © 2014 Pearson Education, Inc. Period • Period is the time interval it takes an object to travel around an entire circular path one time. • Period has units of time, so the SI unit is s. • For constant-speed circular motion, we divide the distance traveled in one period (the circumference of the circular path, 2πr) by the time interval T (its period) to get: • Do not confuse the symbol T for period with the symbol T for the tension force. © 2014 Pearson Education, Inc. Quantitative Exercise 4.3: Singapore hotel • What is your radial acceleration when you sleep in a hotel in Singapore at Earth's equator? • Remember that Earth turns on its axis once every 24 hours, and everything on the planet's surface actually undergoes constant-speed circular motion with a period of 24 hours. © 2014 Pearson Education, Inc. Quantitative Exercise 4.3: Singapore hotel • We want to determine your radial acceleration when you sleep in a hotel in Singapore at Earth's equator. – Earth turns on its axis once every 24 hours, and everything on its surface actually undergoes constant-speed circular motion with a period of 24 hours. • Represent this situation mathematically, THEN solve and evaluate. © 2014 Pearson Education, Inc. Is Earth a noninertial reference frame? • Newton's laws are valid only for observers in inertial reference frames (nonaccelerating observers). – Observers on Earth's surface are accelerating due to Earth's rotation. • Does this mean that Newton's laws do not apply? – The acceleration due to Earth's rotation is much smaller than the accelerations we experience from other types of motion. • In most situations, we can assume that Earth is not rotating and, therefore, does count as an inertial reference frame. © 2014 Pearson Education, Inc. Circular motion component form of Newton's second law © 2014 Pearson Education, Inc. Problem-solving strategy: Processes involving constant-speed circular motion • Sketch and translate – Sketch the situation described in the problem statement. Label it with all relevant information. – Choose a system object and a specific position to analyze its motion. © 2014 Pearson Education, Inc. Problem-solving strategy: Processes involving constant-speed circular motion • Simplify and diagram – Decide if the system can be modeled as a point-like object. – Determine if the constant-speed circular motion approach is appropriate. – Indicate with an arrow the direction of the object's acceleration as it passes the chosen position. – Draw a force diagram for the system object at the instant it passes that position. – On the force diagram, draw an axis in the radial direction toward the center of the circle. © 2014 Pearson Education, Inc. Problem-solving strategy: Processes involving constant-speed circular motion (Cont'd) © 2014 Pearson Education, Inc. Problem-solving strategy: Processes involving constant-circular motion • Represent mathematically – Convert the force diagram into the radial r-component form of Newton's second law. – For objects moving in a horizontal circle (unlike this example), you may also need to apply a vertical y-component form of Newton's second law. © 2014 Pearson Education, Inc. Problem-solving strategy: Processes involving constant-speed circular motion • Solve and evaluate – Solve the equations formulated in the previous two steps. – Evaluate the results to see if they are reasonable (e.g., the magnitude of the answer, its units, limiting cases). © 2014 Pearson Education, Inc. Example 4.5: Toy airplane • A toy airplane flies around in a horizontal circle at constant speed. The airplane is attached to the end of a 46-cm string, which makes a 25° angle relative to the horizontal while the airplane is flying. A scale at the top of the string measures the force that the string exerts on the airplane. • Predict the period of the airplane's motion (the time interval for it to complete one circle). © 2014 Pearson Education, Inc. Example 4.6: Rotor ride • A 62-kg woman is a passenger in a rotor ride. A drum of radius 2.0 m rotates at a period of 1.7 s. When the drum reaches this turning rate, the floor drops away but the woman does not slide down the wall. Imagine that you were one of the engineers who designed this ride. • Which characteristics would ensure that the woman remained stuck to the wall? • Justify your answer quantitatively. © 2014 Pearson Education, Inc. Loop the Loop Consider a ball on a string making a vertical circle. • Draw a free-body diagram of the ball at the top and bottom of the circle • Rank the forces in the two diagrams. Be sure to explain the reasoning behind your rankings • Find the minimum speed of the ball at the top of the circle so that it keeps moving along the circular path • What would happen if the speed was less than the minimum? • What would happen if the speed was more than the miniumum? Slide 6-12 Example Problem: Loop-the-Loop A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight Slide 6-32 Keep the Water in the Bucket Slide 6-12 Roller Coaster and Circular Motion A roller-coaster car has a mass of 500 kg when fully loaded with passengers as shown on the right. 1. If the car has a speed of 20.0 m/s at point A, what is the force exerted by the track at this point? What is the apparent weight of the person? 2. What is the maximum speed the car can have at point B and stay on the track? Slide 15-37 Example 4.7: Texas Motor Speedway • Texas Motor Speedway is a 2.4-km (1.5-mile)-long oval track. One of its turns is about 200 m in radius and is banked at 24° above the horizontal. • How fast would a car have to move so that no friction is needed to prevent it from sliding sideways off the raceway (into the infield or off the track)? © 2014 Pearson Education, Inc. Tip for circular motion • There is no special force that causes the radial acceleration of an object moving at constant speed along a circular path. • This acceleration is caused by all of the forces exerted on the system object by other objects. • Add the radial components of these regular forces. • This sum is what causes the radial acceleration of the system object. © 2014 Pearson Education, Inc. Summary © 2014 Pearson Education, Inc. Summary © 2014 Pearson Education, Inc.