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Gravity Obeys an Inverse-Square Law • Gravity is a universal force that affects all objects in the universe. • Newton proposed that the force of gravity has the following properties: 1. The force is inversely proportional to the square of the distance between the objects. 2. The force is directly proportional to the product of the masses of the two objects. G = 6.67 x 10-11 N.m2/kg2 © 2015 Pearson Education, Inc. Slide 7-1 Gravity on a Grand Scale • No matter how far apart two objects may be, there is a gravitational attraction between them. • Galaxies are held together by gravity. • All of the stars in a galaxy are different distances from the galaxy’s center, and so orbit with different periods. © 2015 Pearson Education, Inc. Slide 7-2 Lecture Presentation Chapter 7 Rotational Motion © 2015 Pearson Education, Inc. Chapter 7 Rotational Motion Chapter Goal: To understand the physics of rotating objects. © 2015 Pearson Education, Inc. Slide 7-4 Chapter 7 Preview Looking Ahead: Rotational Kinematics • The spinning roulette wheel isn’t going anywhere, but it is moving. This is rotational motion. • You’ll learn about angular velocity and other quantities we use to describe rotational motion. © 2015 Pearson Education, Inc. Slide 7-5 Chapter 7 Preview Looking Ahead: Torque • To start something moving, apply a force. To start something rotating, apply a torque, as the sailor is doing to the wheel. • You’ll see that torque depends on how hard you push and also on where you push. A push far from the axle gives a large torque. © 2015 Pearson Education, Inc. Slide 7-6 Chapter 7 Preview Looking Ahead: Rotational Dynamics • The girl pushes on the outside edge of the merry-go-round, gradually increasing its rotation rate. • You’ll learn a version of Newton’s second law for rotational motion and use it to solve problems. © 2015 Pearson Education, Inc. Slide 7-7 Section 7.3 Torque © 2015 Pearson Education, Inc. A question • A force is required for motion. • I: Force causes an acceleration • II: Acceleration changes velocity • III: Velocity changes position • This is true for any translational, circular motions etc. • What about rotational motion? (when the whole object or a system of objects rotates about an axis) © 2015 Pearson Education, Inc. Slide 7-9 Torque • Forces with equal strength will have different effects on a swinging door. • The ability of a force to cause rotation depends on • The magnitude F of the force. • The distance r from the pivot—the axis about which the object can rotate—to the point at which force is applied. • The angle at which force is applied. © 2015 Pearson Education, Inc. Slide 7-10 Torque • Torque (τ) is the rotational equivalent of force. Greek letter Tau • Torque units are newton-meters, abbreviated N ⋅ m. • A force is required to push or pull an object. A torque is required to “turn” an object, i.e., to give a rotational motion to the object. © 2015 Pearson Education, Inc. Slide 7-11 Torque • The radial line is the line starting at the pivot and extending through the point where force is applied. • The angle ϕ (Greek letter: Phi) is measured from the radial line to the direction of the force. © 2015 Pearson Education, Inc. Slide 7-12 Torque • Torque is dependent on the perpendicular component of the force being applied. © 2015 Pearson Education, Inc. Slide 7-13 Torque • The equivalent expression for torque is • For both methods for calculating torque, the resulting expression is the same: • The greater the torque, more effective it would be in rotating the object. © 2015 Pearson Education, Inc. Slide 7-14 How to increase torque • Depends on : • r • F • ϕ © 2015 Pearson Education, Inc. Slide 7-15 QuickCheck 7.10 The four forces shown have the same strength. Which force would be most effective in opening the door? A. B. C. D. E. Force F1 Force F2 Force F3 Force F4 Either F1 or F3 © 2015 Pearson Education, Inc. Slide 7-16 QuickCheck 7.10 The four forces shown have the same strength. Which force would be most effective in opening the door? A. B. C. D. E. Force F1 Force F2 Force F3 Force F4 Either F1 or F3 © 2015 Pearson Education, Inc. Your intuition likely led you to choose F1. The reason is that F1 exerts the largest torque about the hinge. Slide 7-17 Example 7.9 Torque in opening a door Ryan is trying to open a stuck door. He pushes it at a point 0.75 m from the hinges with a 240 N force directed 20° away from being perpendicular to the door. There’s a natural pivot point, the hinges. What torque does Ryan exert? How could he exert more torque? F sin 70° = 226 N. The distance from the hinge to the point at which the force is applied is r = 0.75 m. © 2015 Pearson Education, Inc. Slide 7-18 Example 7.9 Torque in opening a door Ryan is trying to open a stuck door. He pushes it at a point 0.75 m from the hinges with a 240 N force directed 20° away from being perpendicular to the door. There’s a natural pivot point, the hinges. What torque does Ryan exert? How could he exert more torque? In FIGURE 7.20 the radial line is shown drawn from the pivot—the hinge—through the point at which the force is applied. We see that the component of that is perpendicular to the radial line is F⊥ = F cos 20° = 226 N which is also = F sin 70° = 226 N. The distance from the hinge to the point at which the force is applied is r = 0.75 m. PREPARE Problem-solving tips: (1) identify the pivot, (2) identify the radial line (start at the pivot and go to the point where force is applied), (3) determine the angle ϕ. Remember that the angle is measured. © 2015 Pearson Education, Inc. Slide 7-19 Example 7.9 Torque in opening a door (cont.) SOLVE We can find the torque on the door from Equation 7.10: The torque depends on how hard Ryan pushes, where he pushes, and at what angle. If he wants to exert more torque, he could push at a point a bit farther out from the hinge, or he could push exactly perpendicular to the door. Or he could simply push harder! As you’ll see by doing more problems, 170 N ⋅ m is a significant torque, but this makes sense if you are trying to free a stuck door. ASSESS © 2015 Pearson Education, Inc. Slide 7-20 Torque • A torque that tends to rotate the object in a counterclockwise direction is positive, while a torque that tends to rotate the object in a clockwise direction is negative. © 2015 Pearson Education, Inc. Slide 7-21 Net Torque • The net torque is the sum of the torques due to the applied forces: [Insert Figure 7.23] © 2015 Pearson Education, Inc. Slide 7-22 Section 7.4 Gravitational Torque and the Center of Gravity © 2015 Pearson Education, Inc. Gravitational Torque and the Center of Gravity • Gravity pulls downward on every particle that makes up an object (like the gymnast). • Each particle experiences a torque due to the force of gravity. © 2015 Pearson Education, Inc. Slide 7-24 Gravitational Torque and the Center of Gravity • The gravitational torque can be calculated by assuming that the net force of gravity (the object’s weight) acts as a single point. • That single point is called the center of gravity. © 2015 Pearson Education, Inc. Slide 7-25 Example 7.12 The torque on a flagpole A 3.2 kg flagpole extends from a wall at an angle of 25° from the horizontal. Its center of gravity is 1.6 m from the point where the pole is attached to the wall. What is the gravitational torque on the flagpole about the point of attachment? FIGURE 7.26 shows the situation. For the purpose of calculating torque, we can consider the entire weight of the pole as acting at the center of gravity. Because the moment arm r⊥ is simple to visualize here, we’ll use Equation 7.11 for the torque. PREPARE © 2015 Pearson Education, Inc. Slide 7-26 Example 7.12 The torque on a flagpole (cont.) From Figure 7.26, we see that the moment arm is r⊥ = (1.6 m) cos 25° = 1.45 m. Thus the gravitational torque on the flagpole, about the point where it attaches to the wall, is SOLVE We inserted the minus sign because the torque tries to rotate the pole in a clockwise direction. If the pole were attached to the wall by a hinge, the gravitational torque would cause the pole to fall. However, the actual rigid connection provides a counteracting (positive) torque to the pole that prevents this. The net torque is zero. ASSESS © 2015 Pearson Education, Inc. Slide 7-27 Examples in everyday life • You use torque every day without realizing it. You apply torque three times when you simply open a locked door. Turing the key, turning the doorknob, and pushing the door open so it swings on its hinges are all methods of applying a torque. • Athletes • Physical Therapy!!! • Engineering © 2015 Pearson Education, Inc. Slide 7-28 Example 7.16 Angular acceleration of a falling pole In the caber toss, a contest of strength and skill that is part of Scottish games, contestants toss a heavy uniform pole, landing it on its end. A 5.9-m-tall pole with a mass of 79 kg has just landed on its end. It is tipped by 25° from the vertical and is starting to rotate about the end that touches the ground. Estimate the angular acceleration. © 2015 Pearson Education, Inc. Slide 7-29 Example 7.16 Angular acceleration of a falling pole (cont.) PREPARE The situation is shown in FIGURE 7.37, where we define our symbols and list the known information. Two forces are acting on the pole: the pole’s weight which acts at the center of gravity, and the force of the ground on the pole (not shown). This second force exerts no torque because it acts at the axis of rotation. The torque on the pole is thus due only to gravity. From the figure we see that this torque tends to rotate the pole in a counterclockwise direction, so the torque is positive. © 2015 Pearson Education, Inc. Slide 7-30 Example 7.16 Angular acceleration of a falling pole (cont.) ASSESS The final result for the angular acceleration did not depend on the mass, as we might expect given the analogy with free-fall problems. And the final value for the angular acceleration is quite modest. This is reasonable: You can see that the angular acceleration is inversely proportional to the length of the pole, and it’s a long pole. The modest value of angular acceleration is fortunate—the caber is pretty heavy, and folks need some time to get out of the way when it topples! © 2015 Pearson Education, Inc. Slide 7-31 Example 7.18 Starting an airplane engine The engine in a small air-plane is specified to have a torque of 500 N ⋅ m. This engine drives a 2.0-m-long, 40 kg single-blade propeller. On start-up, how long does it take the propeller to reach 2000 rpm? © 2015 Pearson Education, Inc. Slide 7-32 Example 7.18 Starting an airplane engine (cont.) PREPARE The propeller can be modeled as a rod that rotates about its center. The engine exerts a torque on the propeller. FIGURE 7.38 shows the propeller and the rotation axis. © 2015 Pearson Education, Inc. Slide 7-33 Example 7.18 Starting an airplane engine (cont.) ASSESS We’ve assumed a constant angular acceleration, which is reasonable for the first few seconds while the propeller is still turning slowly. Eventually, air resistance and friction will cause opposing torques and the angular acceleration will decrease. At full speed, the negative torque due to air resistance and friction cancels the torque of the engine. Then and the propeller turns at constant angular velocity with no angular acceleration. © 2015 Pearson Education, Inc. Slide 7-34 Why Athletes Need to Understand the Concept of Torque • The concept of torque is a foundation of human movement and is a core principle in physical therapy, personal training, and weightlifting. All movement generates torque to varying degrees and, in reality, it’s what makes the world of biomechanics tick. • You push and pull countless objects in all directions every day in the gym, and the human musculoskeletal system is nothing more than an intricate system of levers and pulleys. While we must generate torque to operate these levers and pulleys, we need to gain awareness of how to minimize torque to avoid injury. Credit: http://breakingmuscle.com/strength-conditioning/why-athletes-need-to-understand-the-concept-of-torque © 2015 Pearson Education, Inc. Slide 7-35 • The resolution of forces occurs when we can: • • Visualize the potential effect of a force on our body. • Determine how much torque we should generate. • Make our muscles and joints act in proportion to this awareness. Credit: http://breakingmuscle.com/strength-conditioning/why-athletes-need-to-understand-the-concept-of-torque © 2015 Pearson Education, Inc. Slide 7-36 • So how can you get your torque on? The answer is simpler than the explanation of torque may lead you to believe: use your muscles. Often in the gym, I see people move into a position without regard for the joint position or joint support. They place ligaments and fascia in charge of holding a joint together (wrong), rather than engaging some of the 642 muscles they have at their disposal (right). Muscles need to do the work of supporting the joint under the strain of torque. Credit: http://breakingmuscle.com/strength-conditioning/why-athletes-need-to-understand-the-concept-of-torque © 2015 Pearson Education, Inc. Slide 7-37 Example: The Snatch A successful snatch is all about torque and how quickly you can move your body around the bar, not the bar around your body. There is a moment at the apex of extension when the sum of all up and down forces are zero. At that point in time, you are exerting no torque, which is precisely how humans are able to lift the amount of weight that we do and avoid the pain and suffering of an annihilating injury Coaches must present the concept of torque to athletes in a consistent and digestible fashion. The health of your joints and your ability to move mass will reflect your understanding of the ability to generate torque. Credit: http://breakingmuscle.com/strength-conditioning/why-athletes-need-to-understand-the-concept-of-torque © 2015 Pearson Education, Inc. Slide 7-38 Everyday See-saw © 2015 Pearson Education, Inc. Lifting an object Slide 7-39 Torque in Automobiles!! • Torque is one of the terms commonly thrown around to describe how powerful a car is, but what exactly does it mean? In a car, torque is applied by the pistons on the crankshaft, causing it and the wheels to turn. © 2015 Pearson Education, Inc. http://study.com/academy/lesson/what-is-torque-definition-equation-calculation.html Slide 7-40 Torque in automobiles • Initial energy that moves a car forward: from thermodynamics. (Example: Internal Combustion engine). • This forces a piston (one or more pistons) down in a straight line, which pushes on a connecting rod and turns the engine's crankshaft. • It's this turning crankshaft where the twisting force of torque initiates. From there the force is carried through a flywheel, transmission, driveshaft, axle(s) and wheel(s) before moving the car. Credit: http://www.edmunds.com/car-technology/the-twist-on-torque.html © 2015 Pearson Education, Inc. Slide 7-41 Summary: General Principles Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-42 Summary: Important Concepts Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-43 Summary: Important Concepts Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-44 Summary: Important Concepts Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-45 Summary Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-46 Summary Text: p. 217 © 2015 Pearson Education, Inc. Slide 7-47