PROBLEM 35E

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem Where θ is the angle that the door is open, I is the moment of inertia of the door about its hinges, b > 0 is a damping constant that varies with the amount of friction on the door, k > 0 is the spring constant associated with the swinging door,θ0 is the initial angle that the door is opened, and V0 is the initial angular velocity imparted to the door (see Figure 4.10). If I and k are fixed, determine for which values of b the door will not continually swing back and forth when closing.

Figure 4.10 Top view of swinging door

Solution

Step 1:

We have given the motion of swinging door with an adjustment screw that controls the amount of friction on the hinges governed by the initial value problem.

Where -moment of inertia of the door about its hinges.

-damping constant that varies with the amount of friction on the door.

initial angle that the door is opened

-initial angular velocity imparted to the door.

and are fixed we have to determine for which values of b the door will not continually swing back and forth when closing.