A massless frame in the shape of a square with 2-m sides has a 1-kg ball at each

Question

A massless frame in the shape of a square with 2-m sides has a 1-kg ball at each corner. Point O is fixed and the axis is perpendicular to the paper. Three forces F1 = 10N on point O makes an angle 60 degree with the vertical side, F2 = 10 N work on mass 1 Kg and makes an angle 30 degree with the vertical side, and the third force of same magnitude works on the diagonally opposite mass to point O and makes an angle 180 degree like shown. What is the angular acceleration (if any)? How much time it takes to cover 6 radians if the system starts from rest? What is the net work done in that time.

Explanation / Answer

torque due to F1 = F1*l*sin60 = 0

torque due to F2 = F2*l2*sin105

l2 = sqrt2*l

torque due to F2 = 10*sqrt2*2*sin105 = 27.32 Nm

torque due to F3 = F3*l*sin90 = 10*2 = -20 Nm

net torque = 27.32 - 20 = 7.32 Nm

moment of inertia about O = I = (1*2^2) + (1*2^2) + (1*(sqrt2*2)^2) = 16 kg m^2

net torque = I*alpha

7.32 = 16*alpha

angular acceleration alpha = 0.4575 rad/s^2    <<<------answer

theta = w*t + (1/2)*alpha*t^2

6 = (1/2)*0.4575*t^2

t = 5.12 s   <<<------answer

work = torqu*displacement

work = 7.32*6 = 43.92 J <<<<------answer

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