Honors Physics Glencoe, 2005 When working in more than one dimension (linear system), we need to revisit some trigonometry Pythagorean Theorem a2+b2=c2 a and b are the sides of a right triangle c is the hypotenuse (or resultant) Law of Cosines R2=a2+b2-2(a)(b)cos(q) a and b are the sides of any triangle q is the angle between a and b R is the resultant Law of Sines a/sin(A) = b/sin(B) = c/sin(C) a, b and c are the sides of any triangle A, B and C are the angles opposite sides a, b and c, respectively A car is driven 125.0km due west, then 65.0km south. What is the magnitude of its displacement? Solve the problem graphically and mathematically and check your answer against each other. An ant is crawling on the sidewalk. At one moment, it is moving south a distance of 5.0mm. It then turns southwest and crawls 4.0mm. What is the magnitude of the ants displacement? Vector Resolution- breaking down into its vertical and horizontal parts SOHCAHTOA sin(q) = opposite/hypotenuse = ay/a cos(q) = adjacent/hypotenuse = ax/a tan(q) = opposite/adjacent = ay/ax Solving for individual sides ax= (a)cos(q) ay= (a)sin(q) If you have more than two vectors, you don’t need to resolve them all before you add them! Use the Tail-To-Tip method and resolve the resultant Angle of Resultant q=tan-1(Ry/Rx) A GPS receiver indicates that your home is 15.0 km and 40.0° N of W, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? Toprameneesha walks 0.40 km in a direction 60.0° W of N, then goes 0.50 km due west. What is his displacement? Kinetic Friction- friction in motion (v>0 m/s) When velocity is constant, a=0 m/s2, therefore Fnet=0 N ▪ Kinetic Friction = Applied Force Static Friction- friction that occurs before the object moves (v=0 m/s) Before an object has moved, the static friction equals the applied force Similar to FN: Ff, static is only as big as it needs to be, to a particular maximum Before touching the couch, Ff, static is o N When applying 1N of force, Ff, static is 1N When applying just enough force to start the couch moving, Ff, static reaches its maximum Ff, static‘s maximum is based on surface irregularities When moving at a constant velocity, the applied force equals the Ff, kinetic Each object has unique surface irregularities, which determines the frictional forces The results of these interactions is the coefficient of static and kinetic friction (ms, mk) The frictional force is based on this coefficient and how hard the surface is holding the object up (FN) Kinetic Friction Ff, kinetic=mkFN Static Friction Ff, static≤msFN See Figure 5-10 A girl exerts a 36N horizontal force as she pulls a 52N sled across a cement sidewalk at a constant speed. What is the coefficient of kinetic friction between the sidewalk and the metal sled runners? Ignore air resistance. You help your mom move a 41kg bookcase to a different place in the living room. If you push with a force of 65N and the bookcase accelerates at 0.12m/s2, what is the coefficient of kinetic friction between the bookcase and the carpet? Equilibrium- when all forces balance/cancel each other out Net force is zero Acceleration is zero Velocity is constant (even if it’s zero) Equilibrant- the additional force needed to cancel out a force or group of forces For two forces acting, it would be the opposite of the resultant of the triangle Find the equilibrant for the following forces. Page 132 There are several forces interacting when working with slopes and incline planes. Fg, FN, Ff, kinetic (or Ff, static) We need to use our trig properties to address these problems. An ant climbs at a steady speed up the side of its anthill, which is inclined at 30° from the vertical. Sketch a free body diagram for the ant. Kohana, who has a mass of 50.0 kg, is at the dentists office having her teeth cleaned. If the only component of her weight perpendicular to the plane of the seat of the chair is 449N, at what angle is the chair tilted?