CHAPTER 5 FORCES IN TWO DIMENSIONS   In this chapter you will:   Represent vector quantities both graphically and algebraically. Use Newton’s Laws to analyze motion when Friction is involved. Use Newton’s Laws and your knowledge of vectors to analyze motion in 2 dimensions.

CHAPTER 5 SECTIONS Section 5.1: Vectors Section 5.2: Friction Section 5.3: Force and Motion in Two Dimensions

SECTION 5.1 VECTORS Objectives Evaluate the sum of 2 or more vectors in 2 dimensions graphically. Determine the components of vectors. Solve for the sum of 2 or more vectors algebraically by adding the components of vectors.

VECTORS REVISITED A vector quantity can be represented by an arrow tipped line segment. -----> Resultant – vector sum of 2 or more vectors.

VECTORS IN MULTIPLE DIMENSIONS The vectors are added by placing the tail of one vector at the head of the other vector.   The resultant is drawn from the tail of the first vector to the head of the last vector. To find the magnitude of the resultant measure its length using the same scale used to draw the 2 vectors. Its direction can be found with a protractor. The direction is expressed as an angle measured counterclockwise from the horizontal.

VECTORS IN MULTIPLE DIMENSIONS In the example in Figure 6-2 we have 95 m east and 55 m north. Our resultant would be 110 m at 30 north of east. In this problem since we get a right triangle we can use a2 + b2 = c2. Force vectors are added in the same way as position or velocity vectors. 90 | II | I 180 -------------|---------------0 , 360 III | IV 270

VECTORS IN MULTIPLE DIMENSIONS The resultant is drawn from the tail of the first vector to the head of the last vector. The angle is found with a protractor. You can Add vectors by placing them Tip-to-Tail (or Head to Tail) and then drawing the Resultant of the vector by connecting the TAIL OF THE FIRST VECTOR TO THE TIP OF THE LAST VECTOR.

VECTORS IN MULTIPLE DIMENSIONS Go Over Figure 5.2 p. 120 Because the length and direction are the only important characteristics of the vector, the vector is unchanged by this movement. This is always true for this type of movement. To find the Resultant you measure it to get its Magnitude and use a Protractor to get its direction. Pythagorean Theorem – if you have a right triangle then you can find the lengths of the sides using a2 + b2 = c2 and here we can use A2 + B2 = R2

VECTORS IN MULTIPLE DIMENSIONS Law of Cosines - The square of the magnitude of the resultant vector is equal to the sum of the magnitude of the squares of the two vectors, minus two times the product of the magnitudes of the vectors, multiplied by the cosine of the angle between them. R2 = A2 + B2 – 2AB cosθ Law of Sines - The magnitude of the resultant, divided by the sine of the angle between two vectors, is equal to the magnitude of one of the vectors divided by the angle between that component vector and the resultant vector. __R__ = __A__ = __B__ sin θ sin a sin b

VECTORS IN MULTIPLE DIMENSIONS Do Example Problem 1 p. 121 A) A2 + B2 = R2 B) R2 = A2 + B2 – 2AB cosθ 152 + 252 = R2 R2 = 152 + 252 – 2(15)(25)cos(135°) 225 + 625 = R2 R2 = 225 + 625 – 750(-.707) 850 = R2 R2 = 850 + 530.33 29.155 km = R R2 = 1380.33 R = 37.153 km   OR Use Components of Vectors

VECTORS IN MULTIPLE DIMENSIONS B) OR Use Components of Vectors  A1x = A cos θ A1y = A sin θ Then A1x = 15 cos(0°) A1y = 15 sin(0°) A2 + B2 = R2 A1x = 15 (1) A1y = 15 (0) (32.68)2 + 17.682 = R2 A1x = 15 A1y = 0 1067.9824 + 312.5824 = R2 A2x = A cos θ A2y = A sin θ 1380.5648 = R2 A2x = 25 cos(45°) A2y = 25 sin(45°) 37.156 km = R A2x = 25 (.707) A2y = 25 (.707) A2x = 17.68 A2y = 17.68 Ax = A1x + A2x Ay = A1y + A2y Ax = 15 + 17.68 Ay = 0 + 17.68 Ax = 32.68 km Ay = 17.68 km Do Practice Problems p. 121 # 1-4

COMPONENTS OF VECTORS Trigonometry – branch of math that deals with the relationships among angles and sides of triangles.   Sine (sin) – opposite side over the hypotenuse; sin  = opp / hyp Cosine (cos) – adjacent side over the hypotenuse; cos  = adj /hyp Tangent (tan) – opposite side over the adjacent side; tan  = opp/adj SOH CAH TOA (sin = opp/hyp ; cos = adj/hyp ; tan = opp/adj

COMPONENTS OF VECTORS We have seen that 2 or more vectors acting in different directions from the same point may be replaced by a single vector, the RESULTANT. The resultant has the same effect as the original vectors   Components of the Vector – the 2 perpendicular vectors that can be used to represent a single vector. You break a vector down into its horizontal and vertical parts. A = Ax + Ay

COMPONENTS OF VECTORS Vector Resolution – the process of breaking a vector into its components. Note the original vector is the hypotenuse of a right triangle and thus larger than both components. Fh = F cos  or Fx = F cos  Fv = F sin  or Fy = F sin  Or Ax = A cos  Ay = A sin 

COMPONENTS OF VECTORS Go Over figure 5.4 p. 122 90   90 Ax < 0 | Ax > 0 Ay > 0 | Ay > 0 II | I 180 --------------|---------------0 , 360 III | IV Ax > 0 | Ax > 0 Ay < 0 | Ay < 0 270

COMPONENTS OF VECTORS If the value for Ay is positive then it moves up and if Ax is positive then it moves to the right. If Ay is negative it moves down and if Ax is negative it moves left.

ALGEBRAIC ADDITION OF VECTORS Angles do not have to be perpendicular in order to use vector resolution.   What you do first is make each vector into its perpendicular components (Components of the Vector). Then the vertical components are added together to produce a single vector that acts in the vertical direction. Next all of the horizontal components are added together to produce a single vector that acts in the horizontal direction.

ALGEBRAIC ADDITION OF VECTORS The resulting vertical and horizontal components can be added together to obtain the Final Resultant. By using Pythagorean Theorem where the Horizontal and Vertical Components are you’re “a” and “b” and you find “R”.   Angle of the Resultant Vector – equals the inverse tangent of the quotient of the y component divided by the x component of the resultant vector. tan  = Ay / Ax or  = tan-1(Ay / Ax)

ALGEBRAIC ADDITION OF VECTORS Note that when tan θ > 0, most calculators give the angle between 0° and 90°, and when tan θ < 0, the angle is reported to be between 0° and −90°. PROBLEM SOLVING STRATEGIES p. 123 Choose a coordinate system (0° always due East) Resolve the vectors into their x-components using Ax = A cos  and their y-components using Ay = A sin  where  is the angle measured Counterclockwise from the positive x-axis. Add or subtract the component vectors in the x-direction. Add or subtract the component vectors in the y-direction. Use the Pythagorean Theorem to find the Magnitude of the resultant vector. To find the Angle of the resultant vector use tan  = Ay / Ax or  = tan-1(Ay / Ax)

ALGEBRAIC ADDITION OF VECTORS Do Example 2 p. 124 A1x = A cos θ A1y = A sin θ Then A1x = 5 cos(90°) A1y = 5 sin(90°) A2 + B2 = R2 A1x = 5 (0) A1y = 5 (1) (-11.49)2 + 4.642 = R2 A1x = 0 A1y = 5 132.0201 + 21.5296 = R2 Ax = A cos θ Ay = A sin θ 153.5497 = R2 Ax = 15 cos(140°) Ay = 15 sin(140°) 12.39 km = R Ax = 15 (-.766) Ay = 15 (.643) Ax = -11.49 Ay = 9.64 tan  = Ay / Ax tan  = 4.64 / -11.49 Ax = A1x + A2x Ay = A1y + A2y tan  = -.4038 -11.49 = 0 + A2x 9.64 = 5 + A2y INV TAN -11.49 = A2x 4.64 = A2y  = -21.99° or 180 – 21.99 = 158.01°

ALGEBRAIC ADDITION OF VECTORS Do Practice Problems p. 125 # 5-10   Do 5.1 Section Review p. 125 # 11-16