EXAMPLES OF EXERCISES. PARALLELOGRAM OF FORCES. 1. State the proposition known as the Parallelogram of Forces. Show that if R be the resultant of two forces P and Q which act in lines meeting in a point, and inclined to each other at an angle i, R2=P2+Q2+2PQ cos i. 2. Given two forces P and Q applied at a point and the angles a and B which the line of the resultant makes with the lines of P and Q respectively: prove from the diagram of the parallelogram of forces that R=P cos a+Q cos B. [In a parallelogram ABDC, let AB represent P, AČ, Q, and AD, R. From B draw BE perpendicular to AD. Then R=AE+ED=P cos a+Q cos B.] 3. Two forces P and Q act in lines inclined to each other at an angle i. R is their resultant, and a and B are the angles which the line of R makes with P and Q respectively. Prove that [In a parallelogram ABDC, let AB represent P, AC, Q, and AD, R. Angle BAC-angle i. From D draw the perpendicular DE to AB or AB produced. AE AB+BE P+Q cos i Then Similarly by drawing DF perpendicular to AC or AC produced, we find AF AC+CF Q+P cos i cos B 4. Two forces 18 poundals and 26 poundals act in directions inclined at an angle of 52° to one another, on a particle. Find the magnitude and direction of their resultant. and [R2=P2+Q2+2PQ cos i =(26)2+(18)2+2 x 26 x 18 x 6157 R-397 poundals 5. If the resultant of two equal forces P, P acting at an angle i be six times as great as if the angle were 2i, show that i=2cos-12 or 2cos-(-3). |