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Consider 4 forces: vector A: 3N, 26° East of North vector B: 5N, 11° West of North vector C: 5/?2 N, 34° West of South vector D: 4N, 64° East of South Then the magnitude of vector A + vector B + vector C + vector D is

Asked by sd8022567 | 09 Jun, 2022, 12:53: PM

Expert Answer

Let

begin mathsize 14px style i with hat on top end style

and

begin mathsize 14px style j with hat on top end style

are unit vectors along +x axis and +y axis respectively

begin mathsize 14px style A with rightwards arrow on top equals space 3 left parenthesis space sin 26 space stack i space with hat on top space plus cos 26 space j with hat on top space right parenthesis space N space equals space left parenthesis space 1.315 space i with hat on top space plus space 2.696 space j with hat on top space right parenthesis space N end style

begin mathsize 14px style B with rightwards arrow on top equals space 5 left parenthesis space minus sin 11 space stack i space with hat on top space plus cos 11 space j with hat on top space right parenthesis space N equals space left parenthesis space minus 0.954 space space i with hat on top space plus space 4.908 space j with hat on top space right parenthesis space N end style

begin mathsize 14px style C with rightwards arrow on top equals space fraction numerator 5 over denominator square root of 2 end fraction left parenthesis negative space sin 34 space stack i space with hat on top space minus cos 34 space j with hat on top space right parenthesis space N equals space left parenthesis space minus 1.977 space i with hat on top space minus space 2.931 space j with hat on top space right parenthesis space N end style

begin mathsize 14px style D with rightwards arrow on top equals space 4 left parenthesis space sin 64 space stack i space with hat on top space minus cos 64 space j with hat on top space right parenthesis space N equals space left parenthesis space 3.595 space i with hat on top space minus space 1.753 space j with hat on top space right parenthesis space N end style

begin mathsize 14px style R with rightwards arrow on top space equals space space A with rightwards arrow on top space plus space B with rightwards arrow on top space plus space C with rightwards arrow on top plus space D with rightwards arrow on top equals space left parenthesis space 1.979 space i with hat on top space plus space 2.920 space j with hat on top space right parenthesis space N end style

Magnitude of above resultant vector is calculated as

begin mathsize 14px style open vertical bar R with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 1.979 right parenthesis squared space plus space left parenthesis 2.920 right parenthesis squared end root space equals space 3.527 space N end style

If direction of

begin mathsize 14px style R with rightwards arrow on top end style

makes angle θ , then tanθ = 2.920 / 1.979 = 1.475 , hence θ = tan^-1 (1.475 ) = 56^o

Above resultant vector can be written as

begin mathsize 14px style A with rightwards arrow on top space plus space B with rightwards arrow on top space plus space C with rightwards arrow on top plus space D with rightwards arrow on top equals space 3.527 space left parenthesis space cos 56 space i with hat on top space plus space sin 56 space j with hat on top space right parenthesis space N end style

Hence resultant vector is of magnitude 3.527 N and direction is 56^o North of East

Answered by Thiyagarajan K | 09 Jun, 2022, 20:28: PM

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