A light shaft along the axis turns steadily in two bearings at

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Rotation, rigid bodies and accelerated frames

Action and Symmetry, Ch The inertia tensor, principal axes and Euler angles

Problem

A light shaft along the

z

axis turns steadily in two bearings at

\omega = 200\ \mathrm{rad\,s^{-1}}

. Two small point masses, each

m = 0.050\ \mathrm{kg}

, are fixed to it, one at

(a, 0, h)

and the other at

(0, a, -h)

, with

a = 0.040\ \mathrm m

and

h = 0.060\ \mathrm m

, in body axes that coincide with the space axes at the instant considered. Clip-on correction weights can be fixed at radius

R = 0.050\ \mathrm m

from the axis in either of two correction planes,

z = +c

and

z = -c

, with

c = 0.080\ \mathrm m

. A weight's angle

\alpha

is measured round the axis from the

+x

direction towards

+y

. ((a)) Find the first moment

\sum m\mathbf r

and the products of inertia

I_{xz}

and

I_{yz}

about the origin, and the size of the bearing torque about the origin. Would a static balancer detect this rotor's imbalance? ((b)) A technician fixes a single weight in the plane

z=+c

, choosing its mass and angle so that the rotor passes the static test. Find that mass and angle, the products of inertia that remain, and the new bearing torque. Does the rotor now run more or less smoothly than before? ((c)) Find the mass and angle of one weight in each correction plane that together make the first moment and both products of inertia vanish.

Browse the problem library