LECTURE 13 CIRCULAR MOTION 3.8 Motion in two dimensions: circular motion What hold you up at the top of a loop- 6.1 Uniform circular motion the-loop? Or do you need to be held up? Velocity and acceleration in uniform circular motion Period, frequency, and speed 6.2 Dynamics of uniform circular motion Maximum walking speed 6.3 Apparent forces in circular motion Centrifugal force? Apparent weight n circular motion Centrifuges Learning objectives 2 ! Relate period, frequency, and speed of an object in a uniform circular motion. ! For an object in a circular motion, relate its acceleration, mass, speed, net force on it, and the radius of the curvature of the path. ! Identify the directions of net force and acceleration of an object in a circular motion. 3.8 Circular motion / 6.1 Velocity and acceleration in uniform circular motion 3 ! The magnitude of centripetal acceleration for uniform circular motion is given by #$ ! = % 6.1 Period, frequency, and speed 4 ! Period, !, is the time interval it takes an object to go around a circle one time. ! Frequency, ", is the number of revolutions per second. 1 " = ! ! The speed of an object in a uniform circular motion is 2'( % = = 2'(" ! Quiz: 6.1-1 5 ! Human centrifuges such as the one shown are used to study the effect of acceleration on the human body and train pilots and astronauts. This one has a radius of 6.1 m and can reach the maximum frequency of 1.1 s!" in as little as around 10 s. ! The image below shows an instance after it is turning at the maximum speed. Draw a vector showing the direction of acceleration for an occupant in the human centrifuge. Top view Quiz: 6.1-1 answer 6 ! Human centrifuges such as the one shown are used to study the effect of acceleration on the human body and train pilots and astronauts. This one has a radius of 6.1 m and can reach the maximum frequency of 1.1 s!" in as little as around 10 s. ! The image below shows an instance after it is turning at the maximum speed. Draw a vector showing the direction of acceleration for an occupant in the human centrifuge. ! The direction of the acceleration of an object in a uniform circular motion is towards the center of the circular path. Top view Quiz: 6.1-2 7 ! Human centrifuges such as the one shown are used to study the effect of acceleration on the human body and train pilots and astronauts. This one has a radius of 6.1 m and can reach the maximum frequency of 1.1 s!" in as little as around 10 s. ! What is the magnitude of the acceleration of the occupant in m/s2 when it is moving at the maximum speed? Quiz: 6.1-2 answer 8 ! Human centrifuges such as the one shown are used to study Limits of tolerance of humans the effect of acceleration on the human body and train pilots and astronauts. This one has a radius of 6.1 m and can reach the maximum frequency of 1.1 s!" in as little as around 10 s. ! What is the magnitude of the acceleration of the occupant in m/s2 when it is moving at the maximum speed? & & % ()'* ( !" ( ! # = = = + 2-. = 6.1 m 2- 1.1 s = ' ' 290 m⁄s( ! This is 308. Humans cannot tolerate this much acceleration for too long. https://en.wikipedia.org/wiki/G-force 6.2 Dynamics of uniform circular motion 9 ! The magnitude of a net centripetal force required for an object with a mass ! going around in a circular path with a radius " with a uniform speed # is given by #* $ = !) = ! %&' " Quiz: 6.2-1 / demo 10 ! A hollow tube lies flat on a table. A ball is shot through the tube. As the ball emerges from the other end, which path does it follow? Quiz: 6.2-1 answer / demo 11 ! A hollow tube lies flat on a table. A ball is shot through the tube. As the ball emerges from the other end, which path does it follow? ! C ! The normal force by the wall of the tube on the ball is the centripetal force necessary for the ball to follow the circular path. ! Without the wall, the ball goes in a straight path due to its inertia. Quiz: 6.2-2 12 ! You are in the front passenger seat of a car. While the car makes a sharp left turn, you found yourself leaning toward the passenger door (to your right). Which of the following statements is/are correct? Choose all that apply. A. Centrifugal force is pushing you to the right. B. Centripetal force is pushing you to the right. C. The seat is exerting a leftward frictional force on you. D. The seat is exerting a rightward frictional force on you. Quiz: 6.2-2 answer 13 ! The seat is exerting a leftward frictional force on you. ! For you to make the turn, there must be centripetal force acting on you toward the center of the curvature. ! The static frictional force by the seat on your bottom is supplying that centripetal force. ! Meanwhile, your head tends to keep going in a straight path due to its inertia. ! This fictitious “force” to the right is sometimes called “centrifugal force.” But there is no such force. Quiz: 6.2-3 14 ! Suppose you are rotating your arm so that your hand is undergoing a vertical uniform circular motion. ! What is the direction of the force your wrist exerts on your hand at the lowest point of the circle? Quiz: 6.2-3 answer 15 ! Suppose you are rotating your arm so that your hand is undergoing a vertical uniform circular motion. ! What is the direction of the force your wrist exerts on your hand at the lowest point of the circle? ! The tension in the wrist on the hand is providing the centripetal force necessary for the hand to go in a circular motion. !"# $%# Quiz: 6.2-4 16 ! Suppose you are rotating your arm so that your hand is undergoing a vertical uniform circular motion. The speed of your hand is 10 m/s (22 mph). Your arm is 0.50 m, and your hand is 0.45 kg. ! What is the magnitude of the force in N your wrist exerts on your hand at the lowest point of the circle? Quiz: 6.2-4 answer 17 ! Suppose you are rotating your arm so that your hand is undergoing a vertical uniform circular motion. The speed of your hand is 10 m/s (22 mph). Your arm is 0.50 m, and your hand is 0.45 kg. ! What is the magnitude of the force in N your wrist exerts on your hand at the lowest point of the circle? ./ ! ! = & − * = ,- = , "#$ '( +( 0 ./ ./ ./ ! & = * + , = ,2 + , = , 2 + '( +( 0 0 0 / > ?@ A/C ! & = 0.45 kg 9.8 m⁄s + = 94 N '( @.D@ A &'( ! If your weight is 500 N, and hang with one arm, the tension in the arm would be 500 N. So 94 N is a reasonable value. *+( Quiz: 6.2-5 18 ! Suppose you are rotating your arm so that your hand is undergoing a vertical uniform circular motion. The speed of your hand is 10 m/s (22 mph). Your arm is 0.50 m, and your hand is 0.45 kg. ! Is the apparent weight of the hand heavier than, lighter than, or equal to the actual weight of the hand at the lowest point of the circle? Quiz: 6.2-5 answer 19 ! The apparent weight of the hand is heavier than the actual weight of the hand. ! The apparent weight of the hand is the contact forces that supports it from free-falling. ! The tension by the wrist on the hand is the apparent weight. ! The tension is greater than the weight of the hand. !"# $%# Quiz: 6.3-1 20 ! You ride a bicycle comfortably around a loop- the-loop. Select all the correct statements when you are at the top of the loop-the-loop. A. There is a gravitational force downward on the bicycle-person system. B. There is a normal force downward on the bicycle- person system. C. The faster you are moving the greater the net force downward. Quiz: 6.3-1 answer / demo 21 ! You ride a bicycle comfortably around a loop-the-loop. Select all the correct statements when you are at the top of the loop-the-loop. A. There is a gravitational force downward on the bicycle-person system. B. There is a normal force downward on the bicycle-person system. C. The faster you are moving the greater the net force downward. - ., *+, ! If your bicycle is touching the track, the normal force on the bicycle is non-zero and downward (normal to the surface). ! The inward force required to make an object move in a circular motion increases '( with increasing speed: ! = & "#$ ) ! Follow-up: What happens if the bicycle is moving very lowly? Quiz: 6.3-1 answer / demo 22 ! The inward force required to make an object move in a circular motion increases '( with increasing speed: ! = & "#$ ) ! Follow-up: What happens if the bicycle is moving very lowly? ! The smaller the speed at the top of the loop-the-loop, the smaller the normal - force, but the smallest it can be is zero. If you do not have enough speed at the ., *+, top (critical speed), the gravitational force will cause you to accelerate more than is needed to go in a circle, and you will come off the track (http://www.youtube.com/watch?v=tzQJNeqiGG4).