FP5

A thin uniform plate (mass M), in the shape of an equilateral triangle (side L), is suspended from one vertex (at A in
the figure), forming a physical pendulum. The triangle swings about an axis perpendicular to the plate through point A. Take x-y coordinates as shown, so that w(y) = y is the width of the triangle a vertical distance y from A.
Our goal is to calculate the period for small oscillations about A.

 

(3 points)  (a)  Find the coordinates of the center of mass ( x_cm , y_cm ). Hint: One method involves
                                breaking the triangle into horizontal rectangular strips of mass dm and then integrating.
                                There is also a symmetry argument.

(4 points)  (b)  Calculate the moment of inertia I_A about the axis through A. Hint: Apply the parallel axis
                                theorem to each horizontal strip and then integrate.

(2 points)  (c)  What is the period for small oscillations about A? Leave your answer in terms of I_A
                                if you were unable to solve part (b).

(2 points)  (d)  ( Extra Credit ) We now move the suspension to a second point B on the y axis such that,
                                when the system is inverted, small oscillations have the same period as about A. Find the
                                coordinates y_B of this point relative to the coordinate system centered on point A.