Physics: Torque. Please answer meticulously A two-way swinging door at a saloon
ID: 1308169 • Letter: P
Question
Physics: Torque. Please answer meticulously
A two-way swinging door at a saloon entrance is made out of a solid plank of wood of mass M and with dimensions shown below. The door hinges are frictionless. To insure the door shuts automatically, it is connected to a torsional spring of spring constant K. This torsional spring is "linear" in that the restoring torque is directly proportional to the angular displacement. Thus the spring follows an angular form of Hooke's law. If the (undamped) door is set into motion and then left alone, what is its angular frequency of oscillations? The saloon owner is getting tired of having the door swing endlessly after someone passes through, and so he decides to install a damping device. The damping device has an adjustable "angular damping coefficient" B, defined by the following relationship found in the device instruction manual: Damping Torque tau D = B omega where omega is the angular velocity of the door about it's hinge. In order to have the door close (and stay closed) in the hortest possible time, what value of B (in terms of other relevant parameters of the problem) should be chosen? Hint: Recall that in class we derived the following solution for the undriven, damped, one-dimensional simple oscillator: x(t) = Aexp[-bt / 2m]cos(cos(omega't + phi), with omega' = [(k/m) - (b / 2m)2]1/2. The system was underdamped for (b / 2m)2 k/m.]Explanation / Answer
Angular frequency = (K/I)^1/2 where I is the moment of inertia of the door w.r.t hinges.
For shortest time, the door must be critically damped (underdamped would culminate into unnecessary oscillations and overdamped would slow the door down)
(B/2I)^2=K/I => B = 2*(KI)^1/2
For a rectangular block I about one edge is M(L^2 + T^2)/3