Question

A 100-kg uniform rectangular plate is supported in the position shown by hinges A and B and by the cable DCE that passes over a frictionless hook at C. Assuming that the tension is the same in both parts of the cable, determine (a) the tension in the cable, (b) the reactions at A and B. Assume that the hinge at B does not exert any axial thrust.

Answer 1

Draw the free body diagram of the plywood:

Find the weight of the plate,

Here, the mass of the plate is and the acceleration due to the gravity is

Substitute for and for

Write the coordinates of each point from the above diagram.

(a)

Let the tension in the cable CE is and the tension in the cable CD is

Consider the moment about line AB,

Here, the unit vector along the line AB is and the weight force vector is

Find the position vector of the line AB

Find the magnitude of the line vector AB,

Calculate the unit vector along the line AB using the relation,

Substitute the known values,

Similarly, find the position vector of the line AC,

Find the position vector of the line AG

Now find the tension vector along CD,,

Here, the unit vector along the line CD is and the magnitude of the tension in cable CD is

Determine the unit vector along the line CD,

Here, the position vector of the line CD is and the magnitude of the vector CD is.

Find the position vector of line CD,

Magnitude of the position vector CD,

Hence, the unit vector along the line CD,

Thus the tension vector along the cable CD,

Write the position vector of line CE,

Find the magnitude of the vector CE,

Find the unit vector along the line CE,

Substitute for and for

Find the tension vector using the relation,

Substitute for,

Now substitute the respected values in equation (1),

Therefore the tension in the cable DCE is (b)

Consider the moment about B with respect to the z-axis:

Hence,

Consider the moment about B with respect to the y-axis:

Substitute for ,

Apply the equilibrium equations,

Substitute for,

Therefore the reaction force vector at point A, is

Resolve the forces along y-direction

Substitute for ,

Resolve the forces along z-direction

Substitute for and for

Therefore, the reaction force vector at B, is