A uniform horizontal shelf of width L and mass m is held to a wall by a frictionless hinge and by a string at an angle of q, measured from the shelf. The string is connected to the shelf at a distance of x from the wall. (a) What is the tension in the string and what are the horizontal and vertical components of the hinge’s force. (8 points) (b) Suppose the string is cut. What is the angular acceleration of the shelf just after the string is cut? (7 points)

The Tension

The horizontal component =

The vertical component =

The angular acceleration ∝ =

Explanation:

The illustration of what the question depicts is shown in the diagram below;

SO; Equilibrium

although the net torque is zero about any point ;

Now;

The horizontal component  is :

The vertical component is :

F⊥ = mg - T sin θ

     =

b) Torque about point 0 =

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west