Question

1.      The x and y components of a certain force are measured and found to be 68

Answer 1

componenents = Fx + e1, Fy + e2

| F | = sqrt ( (Fx + e1)^2 + (Fy + e2)^2 ) = sqrt ( Fx^2 + Fy^2 + 2Fx e1 + 2Fy e2 + e1 ^2 + e2 ^2 )

= sqrt ( Fx^2 + Fy^2 + 2Fx e1 + 2Fy e2)

by binomial expansion,

| F | = sqrt ( Fx^2 + Fy ^2 ) * [ 1 + (Fx e1 + Fy e2) / ( Fx^2 + Fy ^2 ) ] = sqrt ( Fx^2 + Fy ^2 ) + (Fx e1 + Fy e2) / sqrt ( Fx^2 + Fy ^2 )

so, | F | = sqrt ( Fx^2 + Fy^2 ) = 79.9 N = approximately 80 N

error in | F | = (Fx e1 + Fy e2 ) / sqrt ( Fx^2 + Fy ^2 ) = (42 * 2 + 68 * 3) / 80 = 3.6 N

so, | F | = 80 (+-) 3.6 N

tan( angle) = ( Fy + e2) / ( Fx + e1 ) = ( Fy / Fx) [( 1 + e2 / Fy) / (1 + e1/ Fx) ]

= ( Fy / Fx) * [ ( 1 + e2 / Fy) ( 1-e1 / Fx) ] = ( Fy / Fx) ( 1 + ( e2 / Fy - e1 / Fx))

since all these calculations have been shown only for addition of error, but error can be negative also, so for different combinations, we may get e2 / Fy + e1 / Fx instead of e2 / Fy - e1 / Fx. so maximum error is for e2 / Fy + e1 / Fx

so, tan ( angle) = ( Fy / Fx) + (e2 / Fy + e1 / Fx) * ( Fy / Fx)

so, tan ( theta) = Fy / Fx = 0.62

error in tan ( theta) = (e2 / Fy + e1 / Fx) * ( Fy / Fx) = 0.057

Answer 2

Just in case you couldnt read my writing the resultant force is 79.92+/- 3.60N...

and the angle would be 31.7 degrees +/- 0.09 degrees