普通物理 General Physics 6 – Friction, Drag, and Centripetal Force 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 6-1 What Is Physics? 6-2 Friction 6-3 Properties of Friction 6-4 The Drag Force and Terminal Speed 6-5 Uniform Circular Motion 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 6-1 What Is Physics? Describe the frictional force between two objects. Differentiate between static and kinetic friction, study the properties of friction, and introduce the coefficients for static and kinetic friction. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 6-1 What Is Physics? Study the drag force exerted by a fluid on an object moving through the fluid and calculate the terminal speed of the object. Revisit uniform circular motion and using the concept of centripetal force apply Newton’s second law to describe the motion. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 6-2 Friction Static Frictional Force: A force which opposes the tendency of the block to move relative to the surface. In Fig. (a), gravitational force balanced by a normal force . In Fig. (b), a frictional force is balancing . is called the static frictional force. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 6-2 Friction Kinetic Frictional Force: In Fig. (c) to Fig. (e), when the block “breaks away” from its intimate contact with the tabletop and accelerates leftward. The frictional force that then opposes the motion is called the kinetic frictional force. Fig. (g) shows the results of an experiment in which the force on a block was slowly increased until 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 6-2 Friction A frictional force is, the vector sum of many forces acting between the surface atoms of one body and those of another body. The vector sum of many forces acting between the surface atoms of one body and those of another body. In figure, that opposes the motion is the vector sum of the forces at those many chance contacts. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-3 Properties of Friction Property 1. If the body does not move, then the static frictional force and the component of that is parallel to the surface balance each other. They are equal in magnitude, and is directed opposite that component of . F FN mg 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-3 Properties of Friction Property 2. The magnitude of has a maximum value that is given by If the magnitude of the component of that is parallel to the surface exceeds , then the body begins to slide along the surface. μs is the coefficient of static friction. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-3 Properties of Friction Property 3. If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a value given by The constant μk is known as the coefficient of kinetic friction. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-1 If a car’s wheels are “locked” (kept from rolling) during emergency braking, the car slides along the road. Ripped-off bits of tire and small melted sections of road form the “skid marks” that reveal that cold-welding occurred during the slide. The record for the longest skid marks on a public road was reportedly set in 1960 by a Jaguar on the M1 highway in England — the marks were 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-1 290 m long! Assuming that μk= 0.60 and the car’s acceleration was constant during the braking, how fast was the car going when the wheels became locked? 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-1 Solution: By Newton’s second law and 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-2 In Fig. (a), a block of mass m = 3.0 kg slides along a floor while a force of magnitude 12.0 N is applied to it at an upward angle . The coefficient of kinetic friction between the block and the floor is . We can vary from 0 to 90° ( the block remains on the floor ). What gives the 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-2 maximum value of the block’s acceleration magnitude a? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-2 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-3 Although many ingenious schemes have been attributed to the building of the Great Pyramid, the stone blocks were probably hauled up the side of the pyramid by men pulling on ropes. Figure (a) represents a 2000 kg stone block in the process of being pulled up the finished 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-3 (smooth) side of the Great Pyramid, which forms a plane inclined at angle . The block is secured to a wood sled and is pulled by multiple ropes (only one is shown). The sled’s track is lubricated with water to decrease the coefficient of static friction to 0.40. Assume negligible friction at the (lubricated) point where the ropes pass over the edge at the top of the side. If each man 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-3 on top of the pyramid pulls with a (reasonable) force of 686 N, how many men are needed to put the block on the verge of moving? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-3 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-4 A block of mass m1 = 2 kg is placed on a block of mass m2 = 4 kg. The lower block is on a frictionless horizontal surface and is subject to a force F0 = 30 N. Find the minimum value of the coefficient of friction such that m1 does not slide on m2. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-4 Solution: For block 1 For block 2 On adding two terms 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-4 The Drag Force and Terminal Speed (1) Resistance proportional to v: When the velocity of a body relative to a fluid is low, the fluid streams past it in a smooth and continuous way. In such laminar flow, a thin “boundary layer” of fluid forms around the body. The drag force is proportional to the speed: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-4 The Drag Force and Terminal Speed is a constant that depends on the dimensions of the body ( for a sphere is proportional to the radius ) Newton’s second law applied to a body falling vertically through the fluid is 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-4 The Drag Force and Terminal Speed The acceleration drops to zero at the terminal speed , which is given by (2) Resistance proportional to v2: Such as a falling stone or a car moving through the air, the flow of fluid past a body becomes turbulent. The drag force is expressed in 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-4 The Drag Force and Terminal Speed the form where is the density of the fluid, A is the area of the body “projected” onto a plane perpendicular to the motion, and called the drag coefficient. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-4 The Drag Force and Terminal Speed In this case the terminal speed is 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-5 If a falling cat reaches a first terminal speed of 97 km/h while it is tucked in and then stretches out, doubling A, how fast is it falling when it reaches a new terminal speed? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-6 (a) A raindrop with radius R= 1.5 mm falls from a cloud that is at height h = 1200 m above the ground. The drag coefficient C for the drop is 0.60. Assume that the drop is spherical throughout its fall. The density of water is 1000 kg/m3, and the density of air is 1.2 kg/m3, What is the terminal speed of the drop? 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-6 (a) Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-6 (b) What would be the drop’s speed just before impact if there were no drag force? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-5 Uniform Circular Motion A body moves in a circle (or a circular arc) at constant speed v, the body has a centripetal acceleration by “Rounding a curve in a car” or “Orbiting Earth” both are in uniform circular motion, acted on by a centripetal force 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

6-5 Uniform Circular Motion A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the body’s speed. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-7 (a) Igor is a cosmonaut on the International Space Station, in a circular orbit around Earth, at an altitude h of 520 km and with a constant speed v of 7.6 km/s. Igor’s mass m is 79 kg. What is his acceleration? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-7 (b) What force does Earth exert on Igor? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-8 In a 1901 circus performance, Allo “Dare Devil” Diavolo introduced the stunt of riding a bicycle in a loop-the-loop. Assuming that the loop is a circle with radius R = 2.7 m, what is the least speed v Diavolo could have at the top of the loop to remain in contact with it there? 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-8 Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (a) Even some seasoned roller- coaster riders blanch at the thought of riding the Rotor, which is essentially a large, hollow cylinder that is rotated rapidly around its central axis. Before the ride begins, a rider enters the cylinder through a door on the side and stands on a floor, up against a canvas-covered 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (a) wall. The door is closed, and as the cylinder begins to turn, the rider, wall, and floor move in unison. When the rider’s speed reaches some predetermined value, the floor abruptly and alarmingly falls away. The rider does not fall with it but instead is pinned to the wall while the cylinder rotates, as if an unseen (and somewhat unfriendly) agent is pressing the body to the wall. Later, 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (a) the floor is eased back to the rider’s feet, the cylinder slows, and the rider sinks a few centimeters to regain footing on the floor. (Some riders consider all this to be fun.) Suppose that the coefficient of static friction between the rider’s clothing and the canvas is 0.40 and that the cylinder’s radius R is 2.1 m. What minimum speed v must the cylinder and rider have if the rider is not to 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (a) fall when the floor drops? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (a) 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-9 (b) If the rider’s mass is 49 kg, what is the magnitude of the centripetal force on her? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-10 (a) Figure represents a Grand Prix race car of mass m = 600 kg as it travels on a flat track in a circular arc of radius R = 100 m. Because of the shape of the car and the wings on it, the passing air exerts a negative lift downward on the car. The coefficient of static friction between the tires and the track is 0.75. 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-10 (a) If the car is on the verge of sliding out of the turn when its speed is 28.6 m/s, what is the magnitude of ? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-10 (b) The magnitude of the negative lift on a car depends on the square of the car’s speed v2, just as the drag force does. Thus, the negative lift on the car here is greater when the car travels faster, as it does on a straight section of track. What is the magnitude of the negative lift for a speed of 90 m/s? Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-11 Curved portions of highways are always banked (tilted) to prevent cars from sliding off the highway. When a highway is dry, the frictional force between the tires and the road surface may be enough to prevent sliding. When the highway is wet, however, the frictional force may be negligible, and banking is then 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-11 essential. Figure represents a car of mass m as it moves at a constant speed v of 20 m/s around a banked circular track of radius R = 190 m. (It is a normal car, rather than a race car, which means any vertical force from the passing air is negligible.) If the frictional force from the track is negligible, what bank angle prevents sliding? 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-11 Solution: 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-12 A small coin is placed at the rim of a turntable of radius 15 cm which rotates at 30 rev/min. Find the minimum coefficient of friction for the coin to stay on. Solution: Its components are 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 Example 6-12 Since ,we find or 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 6! 2018/11/10 普通物理講義-6 / 國立彰化師範大學物理系/ 郭艷光教授