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Transcript
Chapter 5 Outline Applying Newton’s Laws • Statics • Dynamics • Friction • Kinetic friction • Static friction • Fluid resistance • Circular Motion • Fundamental forces Statics • When a body is not accelerating, we say that it is in equilibrium. • Statics is the study of bodies in equilibrium. • Since the acceleration is zero, we use Newton’s first law to solve these problems. 𝑭=0 𝐹𝑥 = 0; 𝐹𝑦 = 0; 𝐹𝑧 = 0 • Carefully draw a free body diagram that shows all forces acting on the body. • Choose an appropriate coordinate system. Normal Force • When an object is resting on a surface, the surface is exerting a force on the object, called the normal force. • The normal force is always perpendicular to the surface. • The textbook uses the letter 𝑛 for the normal force. It is also common to use 𝐹𝑁 . Statics Example Dynamics • Now we consider a body that is accelerating. • Dynamics is the study of bodies not in equilibrium. • Since the acceleration is not zero, we use Newton’s second law to solve these problems. 𝑭 = 𝑚𝒂 𝐹𝑥 = 𝑚𝑎𝑥 ; 𝐹𝑦 = 𝑚𝑎𝑦 ; 𝐹𝑧 = 𝑚𝑎𝑧 • Carefully draw a free body diagram that shows all forces acting on the body. • Choose an appropriate coordinate system. Apparent Weight • Consider the case of a person standing on a scale in an elevator. • If the elevator is still or moving at a constant velocity, the scale will read the person’s actual weight, 𝑤 = 𝑚𝑔. • If the elevator is acceleration, the scale will read the apparent weight. 𝑛 = 𝑚(𝑔 + 𝑎𝑦 ) • If the body is accelerating downward at the acceleration due to gravity, that is 𝑎𝑦 = −𝑔, it experiences apparent weightlessness. • This is the case during free fall, or while in orbit. Dynamics Example Frictional Forces • So far, we have ignored one of the most important types of force: the force of friction. • Contact force • Parallel to surface Kinetic Friction • When the surfaces are moving relative to each other, we have kinetic friction. 𝑓k = 𝜇k 𝑛 • The force of kinetic friction is proportional to the normal force, 𝑛, and the coefficient of kinetic friction, 𝜇k . • 𝜇k depends on both surfaces. • Does not depend on contact area! Static Friction • When the surfaces are still with respect to each other, we have static friction. 𝑓s ≤ 𝜇s 𝑛 • The force of static friction is proportional to the normal force, 𝑛, and the coefficient of static friction, 𝜇s . • 𝜇s depends on both surfaces. • Generally, 𝜇s > 𝜇k for a given pair of surfaces. Static and Kinetic Friction Coefficient of Rolling Friction • In practice, we often use wheels to reduce the frictional forces involved in moving objects. • • Still, we have to consider the frictional force involved. The coefficient of rolling friction, 𝜇r : 𝑓r = 𝜇r 𝑛 • The frictional force takes into account the deformation of the wheels. • For steel on steel (trains), values of 𝜇r are typically 0.002 to 0.003. Friction Example Fluid Resistance • In the case of the contact between solid surfaces, we find that the frictional force is roughly independent of the relative speed or contact area. • For motion through fluids, the resistive force is quite sensitive to the speed. • For small objects moving at very low speeds, the fluid resistance force, 𝑓, is approximately proportional to the speed. 𝑓 = 𝑘𝑣 • At higher speeds, the fluid resistance force, 𝑓, is approximately proportional to the square of the speed. 𝑓 = 𝐷𝑣 2 Terminal Speed • Since the fluid resistance force is speed dependent, there will be some speed at which the weight of a falling body is balanced by the drag force. • • At this point, the acceleration is zero, and the body has reached its terminal speed. From 𝑓 = 𝐷𝑣 2 , the terminal speed is: 𝑚𝑔 𝑣𝑡 = 𝐷 Fluid Resistance Example Circular Motion • Recall from Chapter 3 that circular motion is produced by a centripetal acceleration towards the center of curvature. 𝑣2 𝑎rad = 𝑅 • The period, 𝑇, is the time for one full circle. 2𝜋𝑅 𝑇= 𝑣 • The acceleration towards the center must be provided by some force. • Planet in orbit: Gravity • Ball on a string: Tension • Car going around turn? Circular Motion Example Consider a racecar going around a turn with a radius of curvature of 250 m. The static and kinetic coefficients of friction between the track and the tires are 𝜇𝑠 = 0.96 and 𝜇𝑘 = 0.68, respectively. • What is the maximum speed at which the car can take the corner without skidding if the track is (a) flat or (b) banked at 18°? Fundamental Forces • We have talked about a lot of different kinds of forces. • Gravitational, friction, normal, fluid resistance, tension… • Are these all actually different in nature? • Four fundamental forces (we think): • Gravitational • Electromagnetic • Strong • Weak • All interactions arise through one of these fundamental forces. • All of the interactions we have dealt with so far were electromagnetic or gravitational. Chapter 5 Outline Applying Newton’s Laws • Statics: 𝑭=0 • Dynamics: 𝑭 = 𝑚𝒂 • Friction • Kinetic friction: 𝑓k = 𝜇k 𝑛 • Static friction: 𝑓s ≤ 𝜇s 𝑛 • Rolling resistance: 𝑓r = 𝜇r 𝑛 • Fluid resistance: 𝑓 = 𝑘𝑣 or 𝑓 = 𝐷𝑣 2 • Terminal velocity: 𝑣𝑡 = • Circular Motion • Fundamental forces 𝑚𝑔 𝐷