Question

A 3.3 kg object is pulled by 4 ropes. One rope is used to pull the object 15° North of East with a force of 25 N. A second rope pulls it due south with a force of 15 N. A third pulls the object 22° west of north with a 36 N magnitude. If the object is moving due west with a constant acceleration (a=1.0 m/s^2 ), describe the force (magnitude and direction) applied by the 4th rope.

Answer 1

mass of the object m = 3.3 kg

Forces F1 = 25 cos 15 i + 25 sin 15 j

                = 24.14 i +6.47 j

Where i , j are the unit vectors along east and north directions respectively

         F2= 15 (-j)        Since it is along south direction

             = -15 j

         F3= 36 sin 22(-i) + 36 cos 22 j        Since direction of force is

             = -13.48 i +33.37 j

Accleration a = 1 m/s ² (-i)    Since it is along west direction

Net force F = ma

                = 3.3 x(-1i)

                = -3.3 i

We know F = F1+F2+F3+F4

From this F4=F-(F1+F2+F3)

                  = -3.3 i -(24.14 i +6.47 j -15 j-13.48 i +33.37 j)

                  = -3.3 i -(10.66 i +24.84 j )

                  = -3.3 i -10.66 i-24.84 j

                   = -13.96 i -24.84 j

So, magnitude of F4 is = [(-13.96) ²+(-24.84) ²]

                                  = 28.49 N

Let F4 makes an angle with west of south then

tan = 24.84 /13.96

           = 1.779

       = tan ^-1(1.779)

          = 60.66 ^o south of west