© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),

© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ), and point of application of force (R).  = R*F(Sin θ) R = Torque Arm = distance between applied force and pivot point Pb.10 – Explain the relationships among speed, velocity, acceleration, and force in rotational systems. http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html

© 2010 Pearson Education, Inc. Circular Motion Circular motion (rotation) - When an object turns about an internal axis - motion of an object in a circle with a constant or uniform speed (velocity???) - constant change in direction = acceleration What is ?

© 2010 Pearson Education, Inc. Period - (cycle) – the time it takes to travel one revolution. -- Object repeatedly finds itself back where it started. Which “dot” (A-D) is traveling the fastest?A B C D Angular Speed (ω) – the rate at which a body rotates around an axis (rotations per minute = rpm)ABCD

© 2010 Pearson Education, Inc. Angular Displacement (θ) - the angle through which a point is rotated SO, which “dot” (A-D) is traveling the fastest? Radial Distance (r) – (radius) – distance from the axis (center) C = 2 π r A B C D

© 2010 Pearson Education, Inc. 5 Circular Motion is characterized by two kinds of speeds: - Tangential (v) (or linear) speed (circumference / t) - Angular speed (ω) (rotational or circular) (θ / t)ABCD Angular speed (ω) = “Clock hand speed”

© 2010 Pearson Education, Inc. Circular Motion—Tangential Speed Tangential speed (symbol v) - The distance traveled by a point on the rotating object divided by the time taken to travel that distance Points closer to the circumference (outer edge) have a higher tangential speed that points closer to the center.

© 2010 Pearson Education, Inc. Circular Motion – Rotational Speed Rotational (angular) speed - (symbol  ) - is the number of rotations or revolutions per unit of time All parts of a rigid merry-go-round or clock hand turn about the axis of rotation in the same amount of time. – all parts have the same rotational speed Tangential speed  Radial Distance  Rotational Speed = r x  

© 2010 Pearson Education, Inc. A ladybug sits halfway between the rotational axis and the outer edge of the turntable. When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge? A. 1 cm/s B. 2 cm/s C. 4 cm/s D. 8 cm/s Rotational and Tangential Speed CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. A ladybug sits halfway between the rotational axis and the outer edge of the turntable. When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge? A. 1 cm/s B. 2 cm/s C. 4 cm/s D. 8 cm/s Rotational and Tangential Speed CHECK YOUR ANSWER --Rotational speed (  ) of both bugs is the same (  20 RPM) --So if the radial distance (r) doubles, tangential speed (v) will also double. --Tangential speed (v) is 2 cm/s  2 = 4 cm/s. = r x  

© 2010 Pearson Education, Inc. Angular Displacement (θ) - the angle through which a point is rotated Radial Distance (r) (radius) - distance from the central axis (center) Angular Speed (ω) – the rate at which a body rotates around an axis (rotations per minute = rpm)

© 2010 Pearson Education, Inc. 11 In circular motion, the direction of any “point” is constantly changing. Therefore, the velocity (speed with direction) is also constantly changing. v1v1 v2v2 A change in direction = A change in velocity = Acceleration a = Δv/t Acceleration only occurs when there is a net force applied to an object a = F/m In which direction must the FORCE be applied in order for an object to continue to travel in constant circular motion? Direction of Force = Direction of Acceleration

© 2010 Pearson Education, Inc. 12 Centripetal Acceleration (A c ) – Acceleration directed toward the center of a circular path --Always points toward center of circle. --(Centripetal = center seeking) --Always changing direction! NOTE: Velocity is always in a straight line.

© 2010 Pearson Education, Inc. Any force directed toward a fixed center is called a centripetal force (F c ). The magnitude of the force required to maintain uniform circular motion. Example: To whirl a tin can at the end of a string, you pull the string toward the center and exert a centripetal force to keep the can moving in a circle. Acceleration only occurs when there is a net force applied to an object a = F/m

© 2010 Pearson Education, Inc. Centripetal Force (F c ) Depends upon: –Mass (m) of object. –Tangential speed (v) of the object. –Radius (r) of the circle. radius 2 mass tangential speed  Centripetal force  Centripetal Force

© 2010 Pearson Education, Inc. Centripetal Force—Example When a car rounds a curve, the centripetal force prevents it from skidding off the road. If the road is wet, or if the car is going too fast, the centripetal force is insufficient to prevent skidding off the road.

© 2010 Pearson Education, Inc. Suppose you double the speed at which you round a bend in the curve, by what factor must the centripetal force change to prevent you from skidding? A. Double B. Four times C. Half D. One-quarter Centripetal Force CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. Suppose you double the speed (v) at which you round a bend in the curve, by what factor must the centripetal force (F c ) change to prevent you from skidding? A. Double B. Four times C. Half D. One-quarter Centripetal Force CHECK YOUR ANSWER Explanation: Because the term for “tangential speed” is squared, if you double the tangential speed, the centripetal force will be double squared, which is four times. The radius is constant. l 2 radius speed tangential mass  forceCentripeta  WARNING: If you make a turn at double the speed, you are 4 times more likely to skid off of the road.

© 2010 Pearson Education, Inc. Suppose you take a sharper turn than before and halve the radius, by what factor will the centripetal force need to change to prevent skidding? A. Double B. Four times C. Half D. One-quarter Centripetal Force CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. l 2 radius speed tangential mass  forceCentripeta  Suppose you take a sharper turn than before and halve the radius; by what factor will the centripetal force need to change to prevent skidding? A. Double B. Four times C. Half D. One-quarter Centripetal Force CHECK YOUR ANSWER Because the term for “radius” is in the denominator, if you halve the radius, the centripetal force will double. WARNING: The sharper your turn (smaller radius), the MORE likely you are to skid. The wider your turn (larger radius), the LESS likely you are to skid. What applies this centripetal force to the car???

© 2010 Pearson Education, Inc. Direction of Centripetal Force (F c ), Acceleration (A c ) and Velocity (v) An object will change direction ONLY if there is a NET force applied to the object. A CONSTANT force applied towards a central point results in a circular path Without a centripetal force, an object in motion continues along a straight-line path. Newton’s 1 st Law

© 2010 Pearson Education, Inc. 22 Direction of Centripetal Force (F c ), Acceleration (A c ) and Velocity (v) Velocity (v) = straight line Force (Fc) = toward center Acceleration (a c ) = toward center Acceleration occurs in the same direction as the applied force

© 2010 Pearson Education, Inc. A tube is been placed upon the table and shaped into a three-quarters circle. A golf ball is pushed into the tube at one end at high speed. The ball rolls through the tube and exits at the opposite end. Describe the path of the golf ball as it exits the tube. N

© 2010 Pearson Education, Inc. 27 What provides the centripetal force? a) Tension b) Gravity c) Friction d) Normal Force Centripetal force is NOT a new “force”. It is simply a way of quantifying the magnitude of the force required to maintain a certain speed around a circular path of a certain radius.

© 2010 Pearson Education, Inc. 28 Tension Can Yield a Centripetal Acceleration: If the person doubles the speed of the airplane, what happens to the tension in the cable? Doubling the speed, quadruples the force (i.e. tension) required to keep the plane in uniform circular motion.

© 2010 Pearson Education, Inc. 30 Centripetal Force: Question Smaller radius: larger force required to keep it in uniform circular motion. A car travels at a constant speed around two curves. Where is the car most likely to skid? Why?

© 2010 Pearson Education, Inc. 31 A.the same as B.one fourth of C.half of D.twice E.four times The answer is E. As the velocity increases the centripetal force required to maintain the circle increases as the square of the speed. Suppose two identical objects go around in horizontal circles of identical diameter but one object goes around the circle twice as fast as the other. The force required to keep the faster object on the circular path is _____ the force required to keep the slower object on the path. radius 2 mass tangential speed  Centripetal force 

© 2010 Pearson Education, Inc. 32 Suppose two identical objects go around in horizontal circles with the same speed. The diameter of one circle is half of the diameter of the other. The force required to keep the object on the smaller circular path is ___ the force required to keep the object on the larger path. A.the same as B.one fourth of C.half of D.twice E.four times The answer is D. The centripetal force needed to maintain the circular motion of an object is inversely proportional to the radius of the circle. Everybody knows that it is harder to navigate a sharp turn than a wide turn. radius 2 mass tangential speed  Centripetal force 

© 2010 Pearson Education, Inc. 33 Suppose two identical objects go around in horizontal circles of identical diameter and speed but one object has twice the mass of the other. The force required to keep the more massive object on the circular path is A.the same as B.one fourth of C.half of D.twice E.four times Answer: D.The mass is directly proportional to centripetal force. radius 2 mass tangential speed  Centripetal force 

© 2010 Pearson Education, Inc. 35 The Normal Force Can Yield a Centripetal Acceleration: How many forces are acting on the car (assuming no friction)? Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

© 2010 Pearson Education, Inc. 36 Banked Curves Why exit ramps in highways are banked? F N cos  = mg F c = F N sin  = mv 2 /r The steeper the bank ( , the less friction is needed (more Force Normal is applied toward the center). This is why/how a race car track can allow cars to travel at such high speeds.

© 2010 Pearson Education, Inc. The ride starts to spin faster and faster and you FEEL yourself being pushed back against the metal cage. Even when the ride starts to tilt skyward and you are looking DOWN at the ground, still you feel yourself almost “forced” back so that you do NOT fall to your death. EXPLAIN….

© 2010 Pearson Education, Inc. Centrifugal Force Although centripetal force is center directed, an occupant inside a rotating system seems to experience an outward force. This apparent outward force is called centrifugal force. Centrifugal means “center-fleeing” or “away from the center.” (The F-word)

© 2010 Pearson Education, Inc. Centrifugal Force – A Common Misconception It is a common misconception that a centrifugal force pulls outward on an object. Example: –If the string breaks, the object doesn’t move radially outward. –It continues along its tangent straight-line path—because no force acts on it. (Newton’s first law) (The F-word)

© 2010 Pearson Education, Inc. An inward net force is required to make a turn in a circle. This inward net force requirement is known as a centripetal force requirement. In the absence of any net force, an object in motion (such as the passenger) continues in motion in a straight line at constant speed. This is Newton's first law of motion. While the car begins to make the turn, the passenger and the seat begin to edge rightward. In a sense, the car is beginning to slide out from under the passenger. Once striking the driver, the passenger can now turn with the car and experience some circle-like motion. There is never any outward force exerted upon the passenger. The passenger is either moving straight ahead in the absence of a force or moving along a circular path in the presence of an inward-directed force. http://www.physicsclassroom.com/mmedia/circmot/rht.cfm

© 2010 Pearson Education, Inc. http://www.physicsclassroom.com/Class/circles/u6l1d2.gif This path is an inward force - a centripetal force. That is spelled c-e-n-t-r-i-p-e-t-a-l, with a "p." The other word - centrifugal, with an "f" - will be considered our forbidden F-word. Simply don't use it and please don't believe in it.

© 2010 Pearson Education, Inc. Rotating Reference Frames Centrifugal force in a rotating reference frame is a force in its own right – as real as any other force, e.g. gravity. Example: – The bug at the bottom of the can experiences a pull toward the bottom of the can.

© 2010 Pearson Education, Inc. Why we use the f-word Centrifugal force in a rotating “reference frame” can be used to examine the simulation of gravity in space stations of the future. By spinning the space station, occupants would experience a centrifugal force (simulated gravity) similar to the bug in the can.

© 2010 Pearson Education, Inc. Simulated Gravity Have a radius of about 1 km (i.e. diameter of 2 km). Rotate at a speed of about 1 revolution per minute. To simulate an acceleration due to gravity, g, which is 10 m/s 2, a space station must:

© 2010 Pearson Education, Inc. Rotational Inertia An object rotating about an axis tends to remain rotating about the same axis at the same rotational speed unless interfered with by some external influence. The property of an object to resist changes in its rotational state of motion is called rotational inertia (symbol I).

© 2010 Pearson Education, Inc. Rotational Inertia Depends upon mass of object. distribution of mass around axis of rotation. –The greater the distance between an object’s mass concentration and the axis, the greater the rotational inertia.

© 2010 Pearson Education, Inc. Rotational Inertia The greater the rotational inertia, the harder it is to change its rotational state. –A tightrope walker carries a long pole that has a high rotational inertia, so it does not easily rotate. –Keeps the tightrope walker stable.

© 2010 Pearson Education, Inc. Rotational Inertia Depends upon the axis around which it rotates Easier to rotate pencil around an axis passing through it. Harder to rotate it around vertical axis passing through center. Hardest to rotate it around vertical axis passing through the end.

© 2010 Pearson Education, Inc. A hoop and a disk are released from the top of an incline at the same time. Which one will reach the bottom first? A. Hoop B. Disk C. Both together D. Not enough information Rotational Inertia CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. A hoop and a disk are released from the top of an incline at the same time. Which one will reach the bottom first? A. Hoop B. Disk C. Both together D. Not enough information Rotational Inertia CHECK YOUR ANSWER Explanation: Hoop has larger rotational inertia, so it will be slower in gaining speed.

© 2010 Pearson Education, Inc. Torque The tendency of a force to cause rotation is called torque. Torque depends upon three factors: –Magnitude of the force –The direction in which it acts –The point at which it is applied on the object

© 2010 Pearson Education, Inc. Torque—Example 1 st picture: Lever arm is less than length of handle because of direction of force. 2 nd picture: Lever arm is equal to length of handle. 3 rd picture: Lever arm is longer than length of handle.

© 2010 Pearson Education, Inc. Suppose the girl on the left suddenly is handed a bag of apples weighing 50 N. Where should she sit order to balance, assuming the boy does not move? A. 1 m from pivot B. 1.5 m from pivot C. 2 m from pivot D. 2.5 m from pivot Rotational Inertia CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. Suppose the girl on the left suddenly is handed a bag of apples weighing 50 N. Where should she sit in order to balance, assuming the boy does not move? A. 1 m from pivot B. 1.5 m from pivot C. 2 m from pivot D. 2.5 m from pivot Rotational Inertia CHECK YOUR ANSWER Explanation: She should exert same torque as before. Torque  lever arm  force  3 m  250 N  750 Nm Torque  new lever arm  force 750 Nm  new lever arm  250N  New lever arm  750 Nm / 250 N  2.5 m

© 2010 Pearson Education, Inc. Center of Mass and Center of Gravity Center of mass is the average position of all the mass that makes up the object. Center of gravity (CG) is the average position of weight distribution. –Since weight and mass are proportional, center of gravity and center of mass usually refer to the same point of an object.

© 2010 Pearson Education, Inc. Center of Mass and Center of Gravity To determine the center of gravity, –suspend the object from a point and draw a vertical line from suspension point. –repeat after suspending from another point. The center of gravity lies where the two lines intersect.

© 2010 Pearson Education, Inc. Center of Gravity—Stability The location of the center of gravity is important for stability. If we draw a line straight down from the center of gravity and it falls inside the base of the object, it is in stable equilibrium; it will balance. If it falls outside the base, it is unstable.

© 2010 Pearson Education, Inc. Angular Momentum The “inertia of rotation” of rotating objects is called angular momentum. momentum. –This is analogous to “inertia of motion”, which was momentum. Angular momentum  rotational inertia  angular velocity –This is analogous to Linear momentum  mass  velocity

© 2010 Pearson Education, Inc. Angular momentum  mass tangential speed  radius Angular Momentum For an object that is small compared with the radial distance to its axis, magnitude of –This is analogous to magnitude of Linear momentum  mass  speed Examples: –Whirling ball at the end of a long string –Planet going around the Sun

© 2010 Pearson Education, Inc. Angular Momentum An external net torque is required to change the angular momentum of an object. Rotational version of Newton’s first law: –An object or system of objects will maintain its angular momentum unless acted upon by an external net torque.

© 2010 Pearson Education, Inc. Suppose you are swirling a can around and suddenly decide to pull the rope in halfway; by what factor would the speed of the can change? A. Double B. Four times C. Half D. One-quarter Angular Momentum CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. Suppose you are swirling a can around and suddenly decide to pull the rope in halfway, by what factor would the speed of the can change? A. Double B. Four times C. Half D. One-quarter Angular Momentum CHECK YOUR ANSWER Explanation: Angular Momentum is proportional to radius of the turn. No external torque acts with inward pull, so angular momentum is conserved. Half radius means speed doubles. Angular momentum  mass tangential speed  radius

© 2010 Pearson Education, Inc. Conservation of Angular Momentum The law of conservation of angular momentum states: If no external net torque acts on a rotating system, the angular momentum of that system remains constant. Analogous to the law of conservation of linear momentum: If no external force acts on a system, the total linear momentum of that system remains constant.

© 2010 Pearson Education, Inc. Suppose by pulling the weights inward, the rotational inertia of the man reduces to half its value. By what factor would his angular velocity change? A. Double B. Three times C. Half D. One-quarter Angular Momentum CHECK YOUR NEIGHBOR

© 2010 Pearson Education, Inc. Angular momentum  rotational inertia  angular velocity Suppose by pulling the weights in, if the rotational inertia of the man decreases to half of his initial rotational inertia, by what factor would his angular velocity change? A. Double B. Three times C. Half D. One-quarter Angular Momentum CHECK YOUR ANSWER Explanation: Angular momentum is proportional to “rotational inertia”. If you halve the rotational inertia, to keep the angular momentum constant, the angular velocity would double.

© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),

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© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),

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Presentation on theme: "© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),"— Presentation transcript:

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