Answer
(a) An upward force of 300 N is required to begin to open the door.
The hinges exert zero force on the door.
(b) An upward force of 150 N is required to begin to open the door.
The hinges exert an upward force of 150 N on the door.
Work Step by Step
(a) Since the force of the hinges on the door is exerted at the axis of rotation, the hinges exert no torque. To open the door, the upward force must exert a torque about the hinges that is at least equal in magnitude to the torque exerted by the weight of the door. Let $L$ be the length of the door. Let $M$ be the mass of the door.
$(\frac{L}{2})F_{up} = \frac{L}{2}(M~g)$
$F_{up} = M~g$
$F_{up} = 300~N$
An upward force of 300 N is required to begin to open the door.
Let $F_h$ be the force exerted on the door by the hinges.
$\sum F = 0$
$F_{up}+F_h - M~g = 0$
$F_h = 300~N - 300~N = 0$
The hinges exert zero force on the door.
(b) We can find the upward force $F_{up}$ when the force is applied at the edge of the door. Let $L$ be the length of the door. Let $M$ be the mass of the door.
$L~F_{up} = \frac{L}{2}(M~g)$
$F_{up} = \frac{M~g}{2}$
$F_{up} = \frac{300~N}{2}$
$F_{up} = 150~N$
An upward force of 150 N is required to begin to open the door.
Let $F_h$ be the force exerted on the door by the hinges.
$\sum F = 0$
$F_{up}+F_h - M~g = 0$
$F_h = 300~N - 150~N$
$F_h = 150~N$
The hinges exert an upward force of 150 N on the door.
