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Sahithyan's S1
Sahithyan's S1 — Mechanics

2D kinematics of a rigid body

Rigid body

A solid body that doesn’t deform.

Degrees of freedom

In the motion of a rigid body in 2D kinematics, there are 33 degrees of freedom.

  • Movement along xx direction
  • Movement along yy direction
  • Rotation about zz direction

In 3D, there are 66 degrees of freedom: movement and rotation along each direction.

Translation

Movement that changes the position of an object. Translation can be done through a rectilinear or curvilinear path. Axes of the body always stays parallel.

Rotation

Circular movement of an object about a fixed axis that is perpendicular to the plane.

General 2D motion

General 2D motion

Mixture of translation and rotation.

vB=vA+vB/A=vA+θ˙×rB/Av_\text{B}= v_\text{A}+v_{\text{B}/\text{A}}= v_\text{A}+\underline{\dot{\theta}}\times r_{\text{B}/\text{A}} aB=aA+aB/A=aA+θ¨×rB/A+θ˙×(θ˙×rB/A)a_\text{B}= a_\text{A}+a_{\text{B}/\text{A}}= a_\text{A}+\underline{\ddot{\theta}}\times r_{\text{B}/\text{A}}+\underline{\dot{\theta}}\times (\underline{\dot{\theta}}\times r_{\text{B}/\text{A}})

Here:

  • θ˙\underline{\dot{\theta}} - Angular velocity of B\text{B} relative to A\text{A}
  • vB/Av_{\text{B}/\text{A}} - Velocity of B\text{B} relative to A\text{A}
  • aB/Aa_{\text{B}/\text{A}} - Acceleration of B\text{B} relative to A\text{A}
  • rB/Ar_{\text{B}/\text{A}} - Position vector of B\text{B} relative to A\text{A}. It’s constant.

In general motion, each particle of the body has a different velocity at every instance.

Instantaneous centre of rotation

The point that has 00 velocity at a particular instant of time. This point might be changing throughout the motion. Denoted by II.

It can be imagined that the object is momentarily having a pure rotation about this centre II.

Finding instantaneous centre of rotation

II can be found by drawing a line perpendicular at the velocity vectors at 2 different points and finding their intersection point.

Centrode

The locus of instantaneous centres during the motion.