  We report measurements of elastic moduli of hcp solid $^4$He down to 15 mK
when the samples are rotated unidirectionally. Recent investigations have
revealed that the elastic behavior of solid $^4$He is dominated by gliding of
dislocations and pinning of them by $^3$He impurities, which move in the solid
like Bloch waves (impuritons). Motivated by the recent controversy of torsional
oscillator studies, we have preformed direct measurements of shear and Young's
moduli of annular solid $^4$He using pairs of quarter-circle shape
piezoelectric transducers (PZTs) while the whole apparatus is rotated with
angular velocity $\Omega$ up to 4 rad/s. We have found that shear modulus $\mu$
is suppressed by rotation below 80 mK, when shear strain applied by PZT exceeds
a critical value, above which $\mu$ decreases because the shear strain unbinds
dislocations from $^3$He impurities. The rotation - induced decrement of $\mu$
at $\Omega = 4$ rad/s is about 14.7 (12.3) % of the total change of temperature
dependent $\mu$ for solid samples of pressure 3.6 (5.4) MPa. The decrements
indicate that the probability of pinning of $^3$He on dislocation segment, $G$,
decreases by several orders of magnitude. We propose that the motion of $^3$He
impuritons under rotation becomes strongly anisotropic by the Coriolis force,
resulting a decrease in $G$ for dislocation lines aligning parallel to the
rotation axis.
