Determine the resultant of the system of forces, Mechanical Engineering

Determine the resultant of the system of forces:

Four forces act on a body as illustrated in Figure. Determine the resultant of the system of forces.

1404_Determine the resultant of the system of forces.png

Figure

Solution

Resolving all of forces along x-axis, we attain

 Rx  = ∑ Fx

= F1 cos θ1 + F2 cos θ2  + F3 cos θ3 + F4 cos θ4

= 40 cos 30o  + 50 cos 315o  + 30 cos 180o  + 20 cos 240o

Note:  The angle created by 50 N forces is measured in anticlockwise direction from positive x axis after making the force work away from O by principle of transmissibility of the force.

∴ Rx  = 40 cos 30o  + 50 cos 45o  - 30 cos 0o  - 20 cos 60o

θ ≤ 90o can be chosen in appropriate quadrant with correct signs as indicated above.

Rx  = 34.64 + 35.36 - 30.00 -10.00

Rx = 30 N

Likewise, resolving all of the forces along y axis, we obtain

Ry = ∑ Fy

= F1 sin θ1 + F2 sin θ2 + F3 sin θ3 + F4 sin θ4

= 40 sin 30o + 50 sin 315o + 30 sin 180o + 20 sin 240o

= 20.00 - 35.36 + 0.00 - 17.32

      Ry = - 32.68 N

Therefore, the resultant in vector form can be expressed as

R¯ = (30 N) i¯ + (- 32.68 N) j¯

The magnitude of the resultant is specified by following

951_Determine the resultant of the system of forces1.png

        = 44.36 N

The direction θ may be worked out as,

θ = tan -1   (Ry /Rx)

= tan -1 (- 32.68/30) = - 47o 26′ 53′′

= 312o 33′ 7′′

The resultant contain a magnitude of 44.36 N and is working in IVth quadrant making an angle of 312o 33′ 7′′ in anticlockwise direction from positive x axis.

Posted Date: 1/28/2013 4:43:26 AM | Location : United States







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