Kinetics of Rigid (Planar) Bodies

Transcription

1 Kinetics of Rigi (Planar) Boies

2 Types of otion Rectilinear translation Curvilinear translation Rotation about a fixe point eneral planar otion

3 Kinetics of a Syste of Particles The center of ass for a syste of particles accelerates in an inertial frae as if it were a single particle with ass (equal to the total ass of the syste) acte upon by a force equal to the net external force. F Fi r& i The rate of change of angular oentu of the syste about a fixe point O is equal to the resultant oent of all external forces acting on the syste about O O M O v O r ir r i The rate of change of angular oentu of the syste about the center of ass is equal to the resultant oent of all external forces acting on the syste about M i ri i Apply these equations to a single rigi boy (really a syste of infinite particles)

4 Rectilinear Translation F Fi r& i v M

5 Rectilinear Translation F Fi r& i v The rigi boy can be treate as a single particle of ass at the CM M o rotation o change in angular oentu o net oent about CM

6 Curvilinear Translation F Fi r& i v M

7 Curvilinear Translation F Fi r& i v The rigi boy can be treate as a single particle of ass at the CM M o rotation o change in angular oentu o net oent about CM

8 Rotation about a fixe point fixe point, O F Fi r& i v M O M O

9 eneral Planar Motion F Fi r& i v M

10 Angular Moentu of a Rigi Boy about its Center of Mass Recall Expression for angular oentu about O O I O ω A siilar erivation I I O boy ( r ) / O ( r ) / boy 2 2 Expression for angular oentu about I ω I usually available in tables it is a property of the rigi boy (unlike I O which also epens on the choice of O)

11 eneral Planar Motion b 2 b θ F Fi r& i v b b 2 M ω θb & I I 3 M ω & & θb I I 3 net oent of all external forces about Mass oent of inertia about Angular acceleration

12 eneral Planar Motion for a Rigi Boy Force balance F F i a i I ( r ) / boy 2 Moent balance M M 2 I αb 3 M a F FBD F 2 IRD

13 Exaple C v 0 A unifor sphere of ass an raius r is projecte along a rough horizontal surface with a linear velocity v 0. The coefficient of kinetic friction between the sphere an the surface is μ k. Deterine: (a) the tie t at which the sphere will start rolling without sliing, an (b) the linear an angular velocities of the sphere at tie t. SOLUTIO:. Draw the FBD with external (incluing reaction) forces on the sphere, an the IRD, an write the three scalar equations. 2. Solve the three corresponing scalar equilibriu equations for the noral reaction fro the surface an the linear an angular accelerations of the sphere. 3. Apply the kineatic relations for uniforly accelerate otion to eterine the tie at which the tangential velocity of the sphere at the surface is zero, i.e., when the sphere stops sliing.

14 Exaple 2 SOLUTIO: k I E E E 4 kg kg. OB 3 kg 2 The portion AOB of the echanis is actuate by gear D an at the instant shown has a clockwise angular velocity of 8 ra/s an a counterclockwise angular acceleration of 40 ra/s 2. Deterine: a) tangential force exerte by gear D, an b) coponents of the reaction at shaft O.. Draw the FBD with external (incluing reaction) forces on AOB, an the IRD, an write the three scalar equations. 2. Evaluate the external forces ue to the weights of gear E an ar OB an the effective forces associate with the angular velocity an acceleration. 3. Solve the three scalar equations erive fro the free-boy-equation for the tangential force at A an the horizontal an vertical coponents of reaction at shaft O.

15 ow to calculate ass oents of inertia? Funaental Result (Parallel Axis Theore) ( ) 2 I I + O r / Mass oent of inertia about any point O Mass oent of inertia about center of ass + Mass oent of inertia about O of a single particle of ass concentrate at the center of ass

16 Free vectors an boun vectors Boun vectors Characterize by an axis Wrong to associate with a single point Free vectors ot characterize by an axis But reference to a point Exaples Exaples Angular velocity of a rigi boy (characterize by an axis) Force applie to a rigi boy (tie to a line of action of the force) Linear velocity of a point P Moent about a point P In general, v P v Q M P M Q

17 Relating oents about two ifferent points F P O P M O M P r O P / O + r F P / O F Q Q M Q M O O / Q + r F or P O M Q M P P / Q P / Q + r F r F Q