What is the total force reactionmagnitude at pin joint O whe

What is the total force reaction(magnitude) at pin joint O when the plate is released from rest?

Solution

The moment the block is released from rest, it will start to rotate about the pin with some angular acceleration.

We will find the net angular acceleration by determining the torque the block would suffer due to the gravitational pull on the block and then using the inertia will get the net angular acceleration

Now for a rectangular sheet, the inertia about the corner O would be given as:

Inertia about the corner = I = M(a² + b²)/3 where a and b are the lengths are breadths of the sheet and M is the mass.

That is, I = 4(0.01 + 0.0025) / 3 = 0.01667 Kg-m²

Also, the net torque about O would Mg(a/2) = 4(9.81)0.05 = 1.962 N-m

Hence, the angular acceleration = Torque / Inertia = 1.962 / 0.01667 = 117.6965 rad/s

Therefore the linear acceleration of the centre of mass of the sheet would be Angular acceleration x Radius

or, a = 117.6965 x distance of O from the COM = 6.579 m/s²

So we can write the equation for the vertical direction as:

Mg - N = Ma where a is the linear acceleration of the block and N is the force being applied by the pin at O.

That is, 4(9.81) - N = 4(6.579)

or, N = 19.3834 N

Therefore the reaction force at O is 19.3834 N