Question

A very thin 2.0-kg disk with a diameter of 80 cm is mounted horizontally to rotate freely about a central vertical axis. On the edge of the disk, sticking out a little, is a small, essentially massless, tab or "catcher." A 2.0-g wad of clay is fired at a speed of 14.0 m/s directly at the tab perpendicular to it and tangent to the disk. The clay sticks to the tab, which is initially at rest, at a distance of 40 cm from the axis.  

(a) What is the moment-of-inertia of the clay about the axis?
kg·m²

(b) What is the moment-of-inertia of the disk about the axis?
kg·m²

(c) What is the moment-of-inertia of both clay and disk about the axis?
kg·m²

(d) What is the linear momentum of the clay before impact?
kg·m/s

(e) What is the angular momentum of the clay with respect to the axis just before impact?
kg·m²/s

(f) What is the angular speed of the disk after impact?
rad/s

Answer 1

moment of inertia of clay about the axis = mass * radius^2

moment of inertia of clay about the axis = 2 * 10^-3 * 0.4^2

moment of inertia of clay about the axis = 0.00032 kg.m^2

moment of inertia of disk = 0.5 * mass * radius^2

moment of inertia of disk = 0.5 * 2 * 0.4^2

moment of inertia of disk = 0.16 kg.m^2

total moment of inertia = 0.00032 + 0.16

total moment of inertia = 0.16032 kg.m^2

linear momentum of clay = mass * velocity

linear momentum of clay = 2 * 10^-3 * 14

linear momentum of clay = 0.028 kg.m/s

angular momentum of clay = mvr

angular momentum = 2 * 10^-3 * 14 * 0.4

angular momentum of clay = 0.0112 kg.m/s

by conservation of momentum

initial momentum = final momentum

0.0112 = 0.16032 * angular speed of disk

angular speed of disk = 0.0698 rad/sec