Question

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)

Answer 1

The Tension
`T = (mgL)/(2xsin \theta)`

The horizontal component
`\sum F_x` =
`(mgL)/(2x) cot \theta`

The vertical component
`\sum F_y` =
`mg(1 - \frac {L}{2x})`

The angular acceleration ∝ =
`(3g)/(2L)`

Step-by-step explanation:

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

SO; Equilibrium

`\sum Fx = 0 = \sum Fy`

although the net torque is zero about any point ;

Now;

`mgL/2 = Tsin \theta .x\\\\Tension \ T = (mgL)/(2x sin \theta)`

The horizontal component is :

`\sum F_x = 0 \\\\F_(11) = T cos \theta \\\\\\= (mgL)/(2x) cot \theta`

The vertical component is :

`\sum F_y = 0`

F⊥ = mg - T sin θ

=
`mg(1 - \frac {L}{2x})`

b) Torque about point 0 =

`mg(L)/(2)= I \alpha \\\\ \alpha = (mgL)/(2I)\\\\\alpha = (mgL)/(2 (mL^2)/(3))\\\\\alpha = (3g)/(2L)`