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types_of_motion_of_a_rigid_body

# types_of_motion_of_a_rigid_body - TYPES OF MOTION OF A...

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Unformatted text preview: TYPES OF MOTION OF A RIGID BODY By deﬁnition a rigid body consists of many mass points (mi) located at different points (r, ) but (r, — 13. ) is ﬁxed so it neither changes shape nor size as it moves. This 151in considbrable simpliﬁcation in describing the two types of motion it can have: _ _ For a rigid body one can deﬁne a center of gravity and show that it is the same as the center of V mass. Zmi 7"; rem : —' ._., 2m!- Consider a rigid body placed some distance above the Earth. i) Each mass m experiences a force Total force on rigid body w = 2W,- = ~Emig3") n as if it was a single object of mass M. ii) Each mass m has potential energy Pg (0 : mg)»,- Total potential energy Pg =zmigyi :ngcm Emigi 2m 1 As if it was a single mass M located at the center ofrnass of the rigid body; Since ycm : TWTESOFRMIHON Translation: All the masses have the same linear velocity and the same linear acceleration “- 2E¢O 21' = A J. V :: V X a [Indeed y = vC,G ] ._.____, ) Rotation about a ﬁxed axis: Now the angular velocity and the angular acceleration are the same for all m ' Now 2F. : 0 For equilibrium we need two conditidns g=0 and g=0 so 2F, 50 27,50 All torques taken about a single axis. The table below summarizes the equations when 9! i 0 and g at 0. (Dynamics) Translation (one dimension, x) ' Rotation (Fixed axis, Z) § 9 g Q. g=a“ g:a",a,=ar,f* —> .R: I] (v, + 6109? Q : (mi + 0502, v, = com" * A 1 A 2g=(x. +v.t+%at2)x @=(®, +w,r+§at2)z v2 =v,2+2a2(x—x,) w2=wi2+2a2(®—®i‘) Displacement along c AS .= PA® M (M333) I = EM”;2 (Moment of Inertia)** M g 2 BF, I Q; = 21', At that point About same axis as I At that time **1 measures the manner in which the mass is distributed around the axis so 27, must also be calculated using the same axis. ...
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