7.2Rotation of a rigid body about afixedaxistyExamplel Theblock moves horizontally ata constantvelocity.The bar OA can berotated around Oaxis,and thebarViskeptclosetothelateraledgeoftheblockasshownintheTA0figure. Given that the height of the block is H, try to find therotational equation, angular velocity and angular acceleration?hof the OA bar.xVotxSoluton: Create rectangularxtgphhcoordinates as shown in the figureβ = arctgTherefore,therotationequationofOAbarishVodp0angularvelocityh? +votdt-2hvitdo&angularaccelerationdt(h? +vit?)?
x y O A 0 v h x Example1 The block moves horizontally at a constant velocity. The bar OA can be rotated around O axis, and the bar is kept close to the lateral edge of the block, as shown in the figure. Given that the height of the block is H, try to find the rotational equation, angular velocity and angular acceleration of the OA bar. Soluton：Create rectangular coordinates as shown in the figure. h v t h x tg 0 = = Therefore, the rotation equation of OA bar is ( ) 0 h v t = arctg angular velocity 2 2 0 2 0 h v t hv dt d + = = angular acceleration 3 0 2 3 2 2 0 2 ( ) d hv t dt h v t − = = + 7.2 Rotation of a rigid body about a fixed axis
7.3Thevelocityaccelerationat each particleofafixed axis rotating rigid bodyWhen the rigid body rotates around a fixed axis, any particleM from the rotationaxis R moves in a circular motion with00O particle as the center and R as the radiuspeMo1.Equation of motion(+)s=r@Mav2. VolecitydsdpThedirectionisMagnitude:Vroshown in figuredtdtThat is, the magnitude of the velocity at any particle in the rotationof the rigidbody is equal to the productof the distance from the particle to the axis ofrotationand theangularvelocityoftherigid body
When the rigid body rotates around a fixed axis, any particle M from the rotation axis R moves in a circular motion with O particle as the center and R as the radius. 1. Equation of motion s = r 2. Volecity The direction is shown in figure. That is, the magnitude of the velocity at any particle in the rotation of the rigid body is equal to the product of the distance from the particle to the axis of rotation and the angular velocity of the rigid body. s (+) O M0 M r v a 7.3 The velocity acceleration at each particle of a fixed axis rotating rigid body Magnitude: r dt d r dt ds v = = =