Chapter 5: Friction, Drag and Elasticity PDF

Chapter 5: Friction, Drag and Elasticity Mrs. Pooja Brahmaiahchari Friction Friction is a force that is around us all the time that opposes relative motion between surfaces in contact but also allows us to move. Friction is parallel to the contact surface between surf...

Chapter 5: Friction, Drag and Elasticity Mrs. Pooja Brahmaiahchari Friction Friction is a force that is around us all the time that opposes relative motion between surfaces in contact but also allows us to move. Friction is parallel to the contact surface between surfaces and always in a direction that opposes motion or attempted motion of the systems relative to each other. If two surfaces are in contact and moving relative to one another, then the friction between them is called Kinetic friction. For example, friction slows a hockey puck sliding on ice. But when objects are stationary, Static friction; The static friction is usually greater than the kinetic friction between the surfaces.  Figure describes a pictorial representation of how friction occurs at the interface between two objects. when you push to get an object moving (in this case, a crate), you must raise the object until it can skip along with just the tips of the surface hitting, break off the points, or do both. A considerable force can be resisted by friction with no apparent motion.  The harder the surfaces are pushed together the more force is needed to move them. Part of the friction is due to adhesive forces between the surface molecules of the two objects. Once an object is moving, there are fewer points of contact, so less force is required to keep the object moving. At small but nonzero speeds, friction is nearly independent of speed. The magnitude of the frictional force has two forms: 1. In static situations (static friction) 2. When there is motion (kinetic friction). Magnitude of static friction is fs is, 𝒇𝒔 ≤ 𝝁𝒔 𝑵, Where 𝑓𝑠 is static friction, 𝜇𝑠 is the coefficient of static friction and N is the magnitude of the normal force. Magnitude of Kinetic friction is fk is, 𝒇𝒌 = 𝝁𝒌 𝑵, 𝜇𝑘 is the coefficient of kinetic friction.  The direction of friction is always opposite that of motion, parallel to the surface between objects, and perpendicular to the normal force. Friction is proportional to the normal force, but not to the area in contact. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area since only high spots touch. When a greater normal force is exerted, the actual contact area increases, and it is found that the friction is proportional to this area. 1. A skier with a mass of 62 kg is sliding down a snowy slope. Find the coefficient of kinetic friction for the skier if friction is known to be 45.0 N. 2. The frictional force between a steel spatula and Teflon frying pan is only 0.200 N. If the coefficient of kinetic friction between two materials is 0.04, calculate the normal force? 3. A box of 15 kg rests on horizontal surface. A) what is the horizontal force that is required to cause the box to begin to slide if the coefficient of static friction is 0.35. B) What is the acceleration of system if person pushes the box with force of 90N if coefficient of kinetic friction is 0.20? 4. A force of 65N is needed to start 8kg box moving across horizontal surface. A) Calculate the coefficient of static friction B)If the box continues to move with an acceleration of 1.4 m/s2. what is the coefficient of kinetic friction. 5. A force of 150N pulls the 30kg box to right as shown below. If coefficient of kinetic friction is 0.25. What is the horizontal acceleration of box? Drag Forces Force of drag on an object when it is moving in a fluid (either a gas or a liquid). You feel the drag force when you move your hand through water. The faster you move your hand, the harder it is to move. Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid.  This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as bicyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force is found to be proportional to the square of the speed of the object. Hence Drag force mathematically written as, 1 𝐹𝐷 = 𝐶𝜌𝐴𝑣 2 2 Where C is the drag coefficient, A is the area of the object facing the fluid, and 𝜌 is the density of the fluid.  Some interesting situations connected to Newton’s second law occur when considering the effects of drag forces upon a moving object. At terminal velocity, 𝐹𝑛𝑒𝑡 = 𝑚𝑔 − 𝐹𝐷 = 𝑚𝑎 = 0 Thus 𝑚𝑔 = 𝐹𝐷 Using the equation of drag force, 1 𝑚𝑔 = 𝐶𝜌𝐴𝑣 2 2 Solving for velocity we obtain, 2𝑚𝑔 𝑣= 𝜌𝐶𝐴  The size of the object that is falling through air presents another interesting application of air drag. The above quadratic dependence of air drag upon velocity does not hold if the object is very small, is going very slow, or is in a denser medium than air. Then we find that the drag force is proportional just to the velocity. This relationship is given by Stokes’law, which states that 𝐹𝑠 = 6𝜋𝑟𝜂𝑣 where r is the radius of the object, 𝜂 is the viscosity of the fluid, and v is the object’s velocity. 6. To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is 0.70 m2) {given: ρ = 1.21 kg/m3and C = 0.28} Elasticity: Stress and strain We now move from forces like friction and drag to those that affects the objects shape. Change in shape due to the application of a force is a deformation. For small deformations, two important characteristics are observed. 1. The object returns to its original shape when the force is removed— that is, the deformation is elastic for small deformations. 2. The size of the deformation is proportional to the force—that is, for small deformations, Hooke’s law is obeyed. In equation form, Hooke’s law is given by, 𝐹 = 𝑘∆𝐿 Where, ∆𝐿 is the amount of deformation, produced by the force F, and k is a proportionality constant that depends on the shape and composition of the object and the direction of the force.  For metals or springs, the straight line region in which Hooke’s law pertains is much larger. Bones are brittle and the elastic region is small and the fracture abrupt. Eventually a large enough stress to the material will cause it to break or fracture. Tensile strength is the breaking stress that will cause permanent deformation or fracture of a material. Changes in Length—Tension and Compression: Elastic Modulus A change in length ∆𝐿 is produced when a force is applied to a wire or rod parallel to its length 𝐿0 , either stretching it (a tension) or compressing it. Tension. The rod is stretched a length when a force is applied parallel to its length. Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, ∆𝐿 is approximately the same for the same magnitude of tension or compression.  For larger deformations, the cross-sectional area changes as the rod is compressed or stretched. ∆𝐿 is proportional to the force and depends on the substance from which the object is made. Additionally, the change in length is proportional to the original length 𝐿0 and inversely proportional to the cross-sectional area of the wire or rod. 1 𝐹 Thus, we get equation: ∆𝐿 = 𝐿0 𝑌 𝐴 Where ∆𝐿 is the change in length, F the applied force, Y is the elastic modulus or Young’s modulus, that depends on the substance, A is the cross-sectional area, and 𝐿0 is the original length. 7. Consider a suspension cable that includes an unsupported span of 3020m. Calculate the amount of stretch in the steel cable (Y = 210 * 109 N/m2). Assume that the cable has a diameter of 5.6 cm and the maximum tension it can withstand is 3.0 x 106 N.  Bones and tendons, which need to be strong as well as elastic, the arteries and lungs need to be very stretchable. The elastic properties of the arteries are essential for blood flow. The pressure in the arteries increases and arterial walls stretch when the blood is pumped out of the heart. The heart is also an organ with special elastic properties. The lungs expand with muscular effort when we breathe in but relax freely and elastically when we breathe out. Our skins are particularly elastic, especially for the young. A young person can go from 100 kg to 60 kg with no visible sag in their skins. The elasticity of all organs reduces with age.  The equation for change in length is traditionally rearranged and written in the following form: 𝐹 ∆𝐿 =𝑌 𝐴 𝐿0 𝐹 The ratio of force to unit area, is defined as STRESS (measured in 𝐴 N/m2). ∆𝐿 The ratio of change in length to length, , is defined as STRAIN. 𝐿0 In other words, Stress = Y Strain Sideways Stress: Shear Modulus Shear deformation behaves similarly to tension and compression and can be described with similar equations. The deformation is called ∆𝑋and it is perpendicular to 𝐿0 , rather than parallel as with tension and compression. The expression for shear deformation is 1𝐹 ∆𝑥 = 𝐿0 𝑆 𝐴 S is the shear modulus,  The spinal column (consisting of 26 vertebral segments separated by discs) provides the main support for the head and upper part of the body. Because the spine is not vertical, the weight of the upper body exerts some of both. Pregnant women and people that are overweight (with large abdomens) need to move their shoulders back to maintain balance, thereby increasing the curvature in their spine and so increasing the shear component of the stress. These higher shear forces increase the risk of back injury through ruptured discs. Changes in Volume: Bulk Modulus An object will be compressed in all directions if inward forces are applied evenly on all its surfaces. Compression is easy in gases but in solids and liquids its extremely difficult. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression.  The relationship of the change in volume to other physical quantities is given by 1 𝐹 ∆𝑉 = 𝑉0 𝐵 𝐴 𝐹 Where B is the bulk modulus, is force per unit area applied uniformly 𝐴 inward on all surfaces, 𝑉0 is the original volume. Note that no bulk moduli are given for gases. One practical example is the manufacture of industrial grade diamonds by compressing carbon with an extremely large force per unit area. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. 8. Suppose you have a 120-kg wooden crate resting on a wood floor. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will its acceleration then be?(given wood on wood µs =0.5 and µk = 0.3) THANK YOU