Question

2. (a) A compound pendulum consists of two coplanar laminar objects that pivot about a
horizontal perpendicular axis. Let M1, I, and d, be the mass, moment of inertia about
a perpendicular axis passing through the center of mass, and perpendicular distance of
the center of mass from the rotation axis, respectively, of the first object. Let M2, 12,
and d be the mass, moment of inertia about a perpendicular axis passing through the
center of mass, and perpendicular distance of the center of mass from the rotation axis,
respectively, of the second object. It is assumed that the point at which the rotation axis
passes through the plane of the two objects and the centers of masses of the two objects
all lie on the same straight line. Demonstrate that the effective length of the pendulum
is
1, + 12 + M,d? + Mod;
L%3D
M, d, + M₂ d₂​

Answer 1

Answer:Transcribed Image Textfrom this Question

(a) A compound pendulum consists of two coplanar laminar objects that pivot about a horizontal perpendicular axis. Let M1, 11, and dj be the mass, moment of inertia about a perpendicular axis passing through the center of mass, and perpendicular distance of the center of mass from the rotation axis, respectively, of the first object. Let M2, 12, and d2 be the mass, moment of inertia about a perpendicular axis passing through the center of mass, and perpendicular distance of the center of mass from the rotation axis, respectively, of the second object. It is assumed that the point at which the rotation axis passes through the plane of the two objects and the centers of masses of the two objects all lie on the same straight line. Demonstrate that the effective length of the pendulum is 11 + 12 + M1d? + M2 dz Mdı + M2d2 (b) Consider a traditional grandfather clock in which the pendulum is made up of a uniform rod of mass M and length 1 with a uniform disk of mass M and radius 1/4 attached to one end (such that the center of the disk corresponds to the end of the rod). The pendulum pivots about the other end of the rod. Demonstrate that the effective length of the pendulum is (131/144)1. L=

Explanation: