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David Raju Vundi 22 May 2017 http://ap-physics.david-s.org Scalars A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: Length Area Volume Time Mass 22 May 2017 http://ap-physics.david-s.org Vectors A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: Displacement Velocity Acceleration Force 22 May 2017 http://ap-physics.david-s.org Vector Diagrams Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction 22 May 2017 http://ap-physics.david-s.org Resultant of Two Vectors The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction: 6N 4N = 10 N = 10 m 6m 4m Vectors in opposite directions: 22 May 2017 6 m s-1 10 m s-1 = 4 m s-1 6N 10 N = 4N http://ap-physics.david-s.org The Parallelogram When Lawtwo vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal 22 May 2017 http://ap-physics.david-s.org When two vectors are joined head to tail Draw the resultant vector by completing the triangle Problem: Resultant of 2 Vectors Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? Solution: Complete the parallelogram (rectangle) The diagonal of the parallelogram ac represents the resultant force The magnitude of the resultant is found using Pythagoras’ Theorem on the triangle abc Magnitude ac 12 5 ac 13 N 2 a 2 12 Direction of ac : tan 5 12 tan 1 67 5 22 May 2017 d 5 b 12 N θ 5N 12 c Resultant displacement is 13 N 67º with the 5 N force http://ap-physics.david-s.org Problem: Resultant of 3 Vectors Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Solution: Find the resultant of the two 5 N forces first (do right angles first) ac 52 52 50 7.07 N 5 tan 1 45 5 Now find the resultant of the 10 N and 7.07 N forces The 2 forces are in a straight line (45º + 135º = 180º) and in opposite directions So, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force 22 May 2017 http://ap-physics.david-s.org 5 d 5N c 5 a 90º θ 45º 135º 5N b Recap What is a scalar quantity? Give 2 examples What is a vector quantity? Give 2 examples How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law? 22 May 2017 http://ap-physics.david-s.org Resolving a Vector Into Perpendicular Components When resolving a vector into Here a vector v is resolved into an x component and a y component y components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other x 22 May 2017 http://ap-physics.david-s.org Practical Applications y=25 N Here we see a table being pulled by a force of 50 N at a 30º angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N 22 May 2017 http://ap-physics.david-s.org 30º x=43.3 N We can see that it would be more efficient to pull the table with a horizontal force of 50 N Calculating the Magnitude of the Perpendicular Components If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are: x = v Cos θ y=v Sin θ y y = v Sin θ θ x=vx Cos θ Proof: x Cos v x vCos 22 May 2017 y Sin v y vSin http://ap-physics.david-s.org Problem: Calculating the magnitude of perpendicular components A force of 15 N acts on a box as shown. What is the horizontal component of the force? Horizontal Component x 15Cos60 7.5 N Vertical Component y 15Sin 60 12.99 N 12.99 Vertical Component N Solution: 60º Horizontal 7.5 Component 22 May 2017 http://ap-physics.david-s.org N Summary If a vector of magnitude v has two perpendicular components x and y, and v makes and angle θ with the x component then the magnitude of the components are: x= v Cos θ y= v Sin θ y=v Sin θ y θ x=v Cosθ 22 May 2017 http://ap-physics.david-s.org