Question

Q1. See the course content for a short review of moment of inertia where you will also find a problem solving video on this topic. . Find the moment of inertia of a uniform thin rod with mass M and length L rotating about its center (a thin rod is a 1D object; in the figure the rod has a thickness for clarity): For this problem, use a coordinate axis with its origin at the rod's center and let the rod extend along the x axis as shown here (in other problems, you will need to generate the diagram): dx dm For a one dimensional object like this rod, we can define linear mass density ( that dm-1dl for a generic length element dl. In this example, dl -dx M/L). We then have Q2 Find the moment of inertia (show your work) of a uniform hoop with mass M and radius R rotating * For a hoop or ring, you can use polar coordinate where dm-ddI-ARdθ where we the arclength

Answer 1

Moment of inertia of rigid body about any axis I_A = Sigma m_1 r^2_i Where r_i = perpendicular distance of m_i from axis. dm = lambda dx dI_A = dm x^2 I_A = integral^+ 1/2_-1/2 lambda dm x^2 = > (lambda x^3/3)^1/2_-1/2 = > lambda L^3/12 M = lambda l lambda = M/2 I_A = M/L times L^3/12 = > ML^2/12 I_A = ML^2/12 \r\n I = integral (R^1)^2 dm dm = M/2 pi d theta R^1 = R cos theta I = integral^2 pi_0 R^2 cos^2 theta d theta = MR^2/2 pi integral^2 pi_0 cos^2 theta d theta I = MR^2/2 pi [theta/2 + sin 2 theta/2] Vertical-bar ^2 pi_0 I = MR^2/2 pi [pi] I = MR^2/2