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8. Dynamics of a rigid body

8. Dynamics of a rigid body. Theorems. r CM : location of the center of mass referred to an inertial frame /Oxyz v CM : velocity of the center of mass /Oxyz P , L : total momentum, angular momentum of the system /Oxyz

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8. Dynamics of a rigid body

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  1. 8. Dynamics of a rigid body • Theorems rCM : location of the center of mass referred to an inertial frame /Oxyz vCM : velocity of the center of mass /Oxyz P , L : total momentum, angular momentum of the system /Oxyz P/CM , L/CM : momentum, angular momentum of the system referred to the center of mass Fext,text : external force, torque, M: total mass of the system B. Rossetto

  2. 8. Dynamics of a rigid body • Translation and rotation (Bold letter are vectors) Translation Rotation B. Rossetto

  3. 8. Moment of inertia • Definition G • Theorem O Proof. B. Rossetto

  4. 8. Moment of inertia • Homogeneous sphere/z’Oz z r sin j Radius: R, mass: M, density : r Contribution of the element dm, length: r sinj dq, weidth: r dj height : dr,, distance: r sinj j r y 0 q x r sin j r dj r r sin jdq j 0 B. Rossetto

  5. 8. Dynamics of a rigid body • Variable mass system Example: rocket vertical motion. If -dm is the positive value of the mass of expelled gases, v and v’ the rocket and gas exhaust velocities relative to earth axes, the total momentum at t+dt is: (m+dm)(v+dv)+(-dm)v’ =mv+mdv-v0dm, with v0=v’-v Momentum conservation requires that it is equal to momentum at t: mv+mdv-v0dm=mv From the second law: and then: B. Rossetto

  6. 8. Dynamics of a rigid body • Gyroscopic precession Fundamental theorem . Torque of the weight: . G and Variation of the angular momentum O The axis of rotation of the gyroscope, given by the direction of the angular momentum, turns aroud a vertical axis, parallel to weight Exercice. Find the precession velocity B. Rossetto

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