1 / 12

Dynamics of Rotational Motion

Dynamics of Rotational Motion. The main problem of dynamics: How a net force affects (i) translational (linear) motion ( Newtons’ 2 nd law) (ii) rotational motion ??? (iii) combination of translational and rotational motions ???. m. α z.

chambray-braun
Télécharger la présentation

Dynamics of Rotational Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dynamics of Rotational Motion The main problem of dynamics: How a net force affects (i) translational (linear) motion (Newtons’ 2nd law) (ii) rotational motion ??? (iii) combination of translational and rotational motions ??? m αz Lever arm lis the distance between the line of action and the axis of rotation, measured on a line that is to both. Axis of rotation Definition of torque: τz > 0 if the force acts counterclockwise τz < 0 if the force acts clockwise Units: [ τ ] = newton·meter = N·m A torque applied to a door

  2. Newton’s Second Law for Rotation about a Fixed Axis Only Ftan contributes to the torque τz . • One particle moving on a circle: Ftan=matan and atan= rαz rFtan= mr2 αzτz = I αz τzI (ii) Rigid body (composed of many particles m1, m2, …) Example 10.3: a1x,T1,T2? a1x Pulley: T2R-T1R=Iαz T2-T1=(I/R2)a1x y Glider: T1=m1a1x a1x=a2y=Rαz a2y Object: m2g-T2=m2a1x Only external torques (forces) count ! X

  3. Work-Energy Theorem and Power in Rotational Motion Rotational work: Work-Energy Theorem for Rigid-Body Rotation: Proof: Power for rotational work or energy change:

  4. Rigid-Body Rotation about a Moving Axis General Theorem: Motion of a rigid body is always a combination of translation of the center of mass and rotation about the center of mass. Energy: Proof: General Work-Energy Theorem: E – E0 = Wnc , E = K + U Rolling without slipping: vcm= Rω, ax = R αz

  5. Rolling Friction Rolling Motion Sliding and deformation of a tire also cause rolling friction.

  6. Combined Translation and Rotation: Dynamics Note: The last equation is valid only if the axis through the center of mass is an axis of symmetry and does not change direction. Exam Example 24: Yo-Yo has Icm=MR2/2 and rolls down with ay=Rαz (examples 10.4, 10.6; problems 10.20, 10.75) Find: (a) ay, (b) vcm, (c)T Mg-T=May τz=TR=Icmαz ay=2g/3 , T=Mg/3 y ay

  7. Exam Example 25: Race of Rolling Bodies(examples 10.5, 10.7; problem 10.22, problem 10.29) y Data: Icm=cMR2, h, β Find: v, a, t, and min μs preventing from slipping FN x = h / sinβ β x Solution 1: Conservation of Energy Solution 2: Dynamics (Newton’s 2nd law) and rolling kinematics a=Rαz fs v2=2ax

  8. Angular Momentum (i) One particle: Impulse-Momentum Theorem for Rotation (ii) Any System of Particles: (nonrigid or rigid bodies) It is Newton’s 2nd law for arbitrary rotation. Note: Only external torques count since Unbalanced wheel: torque of friction in bearings. (iii) Rigid body rotating around a symmetry axis:

  9. Principle of Conservation of Angular Momentum Total angular momentum of a system is constant (conserved), if the net external torque acting on the system is zero: For a body rotating around a symmetry axis: I1ω1z = I2ω2z Example: Angular acceleration due to sudden decrease of the moment of inertia ω0 < ωf Origin of Principles of Conservation There are only three general principles of conservation (of energy, momentum, and angular momentum) and they are consequences of the symmetry of space-time (homogeneity of time and space and isotropy of space).

  10. Hinged board (faster than free fall) Ball: m Board: I=(1/3)ML2 L h=L sinα Mg

  11. Gyroscopes and Precession Precession is a circular motion of the axis due to spin motion of the flywheel about axis. Period of earth’s precession is 26,000 years. Dynamics of precession: Precession angular speed: Circular motion of the center of mass requires a centripetal force Fc = M Ω2 r supplied by the pivot. Nutation is an up-and-down wobble of flywheel axis that’s superimposed on the precession motion if Ω≥ω.

  12. Analogy between Rotational and Translational Motions

More Related