Methods of Finding Vector Sum Phys 13 General Physics 1 Methods of Finding Vector Sum MARLON FLORES SACEDON

Vector analysis Graphical Method Analytical Method Methods of finding Vector sum or resultant of forces Parallelogram Method Polygon Method Graphical Method Cosine law Method Component Method Analytical Method

Vector analysis Two forces Graphical Method Parallelogram Method 𝑅 𝜃 𝛽 𝛽 𝐹 2 𝐹′ 2 𝜃 𝐹 1 𝐹′ 1 𝜃 𝛽 Two forces (Drawn to scale)

Vector analysis Two forces Graphical Method Polygon Method 𝐹 2 𝐹 1 𝜃 𝛽 Two forces (Drawn to scale)

Vector analysis Two forces Graphical Method Polygon Method 𝑅 𝐹 2 𝐹 2 𝛽 𝜃 𝐹 1 𝐹 1 𝜃 𝛽 𝜃 Two forces (Drawn to scale) This is Polygon method

Vector analysis Two forces Analytical Method Cosine law Method 𝑅 𝐹 2 𝜃 𝛽 𝐹 2 𝐹 1 Equivalent polygon of vectors (Drawn not to scale)) 𝐹 1 𝛽 𝜃 𝐹 2 Two forces Drawn not to scale Apply law sines to solve for 𝛼: 𝑠𝑖𝑛𝛼 𝐹 2 = 𝑠𝑖𝑛 𝜃+𝛽 𝑅 𝛼 𝛾 Magnitude of the Resultant R (apply law of cosines) 𝑅 or 𝑅= 𝐹 1 2 + 𝐹 2 2 −2 𝐹 1 𝐹 2 cos⁡(𝜃+𝛽) Direction of the Resultant R 𝛾= 180 𝑜 −𝜃−𝛼

Vector analysis Analytical Method Component Method Vector can be resolve in x & y components + 𝐴 𝑦 𝐴 𝜃 + 𝐴 𝑥

Vector analysis Analytical Method Component Method Vector can be resolve in x & y components Components and its vector can be formed into a right triangle. + 𝐴 𝑦 𝜃 𝐴 + 𝐴 𝑦 𝐴 𝑨 = 𝑨 𝒙 + 𝑨 𝒚 Resolution of vectors Composition of vectors 𝜃 + 𝐴 𝑥 + 𝐴 𝑥 From right triangle (left side figure) 𝐴= 𝐴 𝑥 2 + 𝐴 𝑦 2 𝑐𝑜𝑠𝜃= 𝐴 𝑥 𝐴 𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃 𝜃= 𝑇𝑎𝑛 −1 𝐴 𝑦 𝐴 𝑥 𝑠𝑖𝑛𝜃= 𝐴 𝑦 𝐴 𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃

Vector analysis Two forces or more Analytical Method Component Method Steps in Component Method Step 2: Sum up all components along x-axis and all components along y-axis algebraically. Step 1: Resolve the vectors into each components and solve their values using 𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃 & 𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃. 𝐹 𝑖𝑥 = 𝐹 𝑖 𝑐𝑜𝑠𝜃 From the Figure (left) we have, 𝛽 𝐹 2 + 𝐹 2𝑦 + 𝐹 1𝑦 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +… 𝐹 1 𝜃 𝐹 𝑖𝑦 = 𝐹 𝑖 𝑠𝑖𝑛𝜃 Components of Resultant 𝑅 along x & y axis 𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +… Step 3: Calculate the magnitude and direction of the resultant using 𝑅 𝑥 and 𝑅 𝑦 . − 𝐹 1𝑥 𝐹 2𝑥 𝑅= Σ𝑅 𝑥 2 + Σ𝑅 𝑦 2 Magnitude of resultant Two forces or more Drawn not to scale 𝛾= 𝑇𝑎𝑛 −1 𝑅 𝑦 𝑅 𝑥 direction of resultant

Vector analysis Steps in Component Method + 𝐹 1𝑦 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +… Step 1: Resolve the vectors into each components and solve their values using 𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃 & 𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃. Step 2: Sum up all components along x-axis and all components along y-axis algebraically. From the Figure (left) we have, 𝛽 𝐹 2 + 𝐹 1𝑦 𝑹 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +… 𝐹 1 𝜃 𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +… Step 3: Calculate the magnitude and direction of the resultant using 𝑅 𝑥 and 𝑅 𝑦 . − 𝐹 1𝑥 + 𝐹 2𝑦 𝑅= Σ𝑅 𝑥 2 + Σ𝑅 𝑥 2 Magnitude of resultant 𝛾= 𝑇𝑎𝑛 −1 𝑅 𝑦 𝑅 𝑥 direction of resultant 𝛾 𝐹 2𝑥

Vector analysis Problem Find the magnitude and direction of resultant using polygon method, cosine law and component method. 200𝑔𝑓 30 𝑜 65 𝑜 500𝑔𝑓 35 𝑜 400𝑔𝑓

Vector analysis 𝜸 Application Problem end Displacement (D) 𝐷 𝑦 8 km Application Problem A cross-country skier skis 5.00 km in the direction 500 east of south, then 3.00 km in the direction N 600 E, and finally 8.00 km with bearing angle of 3380. Find the displacement of the skier. N E S W 𝜸 Solving for the x component of displacement =2.82 𝑘𝑚 𝐷 𝑥 start 𝐷 𝑥 = +5 cos 50 𝑜 50o + 3 cos 30 𝑜 −8 cos 68 𝑜 5 km Solving for the y component of displacement N E S W 𝐷 𝑦 = − 5 sin 50 𝑜 + 3 sin 30 𝑜 + 8 sin 68 𝑜 =5.09 𝑘𝑚 68o 338o Solving for the magnitude and direction of displacement 3 km N E S W 60o 𝐷= 2.82 2 + 5.09 2 =5.82 𝑘𝑚 30o 𝛾= 𝑇𝑎𝑛 −1 5.09 2.82 = 61 𝑜 East of North

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