Question

C11B.8

b towaiu yö at a speed (a) What is the magnitude and direction of the pipe's angular velocity vector (b) What is the pipe's rotational energy? (c) What is its total energy K+U? CI C11B.8 Calculate the total energy K Ut of a rod of mass M and length L constrained to rotate with an angular velocity of around one of its ends. (Hint: Treat the rod as an object whose center of mass is moving and which is also rotating around that center of mass. How fast does the rod rotate around its center of mass in this situation?) C11 Modeling C11M.1 Calculate the moment of inertia of a wheel of total mass M and radius R constructed as shown in the figure

Answer 1

The mass of the rod is m. the center of mass of the red is at distance L/2 from the end the red is rotating about rightarrow the velocity of the center of mass of rod: V = (distance of cm from rotating end) times angular velocity V = L/2 W V_cm = LW/2 so translational K E = 1/2 M V_cm^2 = 1/2 M (LW/2)^2 K = 1/8 M L^2 W^2 \r\n The rotational kinetic energy: There might be two ways to calculate the rotational K E since the rod is rotating about end so: moment of inertia of rod about its are end = 1/12 ML^2 therefore Rotating with angular velocity W so V^rot = 1/2 I w^2 = 1/24 ML^2 W^2 V^rot = 1/24 ML^2w^2 \r\n the moment of inertia (I) of rod about its COM = 1/3 ML^2 therefore when the rod rotating with angular velocity w about its end