Question

Q4 A ring of mass M and radius R is attached so that it can pivot asone of a rod of mass M and length 2R. the system is suspended by a pin at the other end of the rod and all in a uniform gravitational field,g; thus forming a double pendulum. The motion is restricted to a plane A) Show that the kinetic energy and potential energy can be written for small oscillations as: B) Find the natural frequencies for small oscillations. C) Find the normal modes for small oscillations. Give a physical description of pure mode oscillations. fn rigid body around its middle

Answer 1

Hence K.E of system K.E = 1/2 I_rod theta^2_1 + 1/2 m v^2 net because theta^rightarrow_net = v^rightarrow_1 + v^rightarrow_1 + v^rightarrow_2 rightdoublearrow because I_rod = M (2R)^2/3 v^2_net = 4 R^2 theta^2_1 + R^2 theta^2_2 + 4 R^2 theta^2_2 + 4 R^2 theta_1 theta_2 cos (theta_2 - v_1) K.E = 1/2 (4 MR^2/3) theta^2_1 + 1/2 M (4R^2 theta^2_1) + 1/2 MR^2 theta^2_2 + 1/2 M [4R^2 theta_1 theta_2 cos (theta_2 - theta_1)] [K.E = 1/2 M [16/3 theta^2_1 + theta^2_2 + 4 theta_1 theta_2 cos (tehta_2 - theta_1)] R^2 For small oscillation theta_2 - theta_1 small cos (theta_2 - theta_1) almostequalto 1 K.E = 1/2 MR^2 [16/3 theta^2_1 + theta^2_2 + 4 theta_1 theta_2] \r\n P.E of system P.E = -mg R cos theta_1 - Mg (2R cos theta_1) - Mg R cos theta_2 P.E = -mg R (3 cos theta_1 + cos theta_2) P.E = -Mg R (3 cos theta_1 + cos theta_2) For theta_1 and theta_2 small