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Summary

  • Ideally, a rigid body is one for which the distances between different particles of the body do not change, even though there are forces on them.
  • The centre of mass of a uniform sphere is at its geometrical centre. 
  • For a thin rod of a uniform cross-section and density, the centre of mass is at its geometrical centre. 
  • For a thin circular plane ring, the centre of mass is again at its geometrical centre where there is actually no matter. This is an example of a body whose centre of mass lies outside the body. 
  • A rigid body fixed at one point or along a line can have only rotational motion. A rigid body not fixed in some way can have either pure translation or a combination of translation and rotation.
  • In rotation about a fixed axis, every particle of the rigid body moves in a circles which lies in a plane perpendicular to the axis and has its center on the axis. Every point in the rotating rigid body has the same angular velocity at any instant of time.
  • In pure translation, every particle of the body moves with the same velocity at any instant of time.
  • Angular velocity is a vector. Its magnitude is ω = dθ/dt and it is directed along the axis of rotation. For rotation about a fixed axis, this vector ω has a fixed direction.
  • The vector or cross product of two vector a and b is a vector written as a x b. The magnitude of this vector is absinθ and its direction is given by the right handed screw or the right hand rule.
  • The linear velocity of a particle of a rigid body roating about a fixed axis is given by v = ω x r, where r is the position vector of the particle with respect to an origin along the fixed axis. The relation applies even to more general rotation of a rigid body with one fixed point. In that case r is the position vector of the particle with respect to the fixed point taken as the origin.
  • The centre of mass of a system of particles is defined as the point whose position vector is
  • Velocity of the centre of mass of a system of particles is given by V =P/M, where P is the linear momentum of the system. The centre of mass moves as if all the mass of the system is concentrated at this point and all the external forces act at it. If the total external force on the system is zero, the the total linear momentum of the system is constant.
  • The anglar momentum of a system of n particles about the origin is
  • The torque or momentum of a system of n particles about the origin is  
  • The force Fi acting on the ith particle includes the external as well as internal forces. Assuming Newton's third law and that forces between any two particles act along the line joining the particles, we can show ζint=0 and
  • A rigid body is in mechanical equilbrium if
  • it is in translation equilibrium, i.e., the total external force on it is zero : ,and
  • It is in rotational equilibrium, i.e.the total external torque on it is zero,
  • The centre of gravity of an extended body is that point where the total gravitational torque on the body is zero.
  • The moment of inertia of a rigid body about an axis is defined by the formula where ri is the perpendicular distance of the i th point of the body from the axis. The kinetic energy of rotation is
  • The theorem of parallel axes: allows us to determine the moment of inertia of a rigid body about an axis as the sum of the moment of inertia of the body about a parallel axis through its centre of mass and the product of mass and square of the perpendicular distance between these two axes.
  • Rotation about a fixed axis is dierctly analogous to linear motion in respect of kinematics and dynamics.
  • The angular acceleration of a rigid body rotating about a fixed axis is given by . If the external torque is zero, the component of angular momentum about a fixed axis of such a rotating body is constant.
  • For rolling moton with out slipping , where is the velocity of translation (i.e of the centre of mass), r is the radius and m is the mass of the body. The kinectic energy of such a rolling body is the sum of kinectic energies of translation and rotation:
  • Separating the motion of a system of particles as, i.e. the motion of the centre of mass transnational motion of the system and motion about (i.e. relative to) the centre of mass of the system is a useful technique in dynamics of a system of particles. One example of this technique is separating the kinetic energy of a system of particles K as the kinetic energy of the system about its centre of mass K′ and the kinetic energy of the centre of mass MV2/2, K = K ′ + MV2/2
  • Newton's Second Law for finite sized bodies (or systems of particles) is based in Newton's Second Law and also Newton's Third Law for particles. 
  • The centre of gravity of a body coincides with its centre of mass only if the gravitational field does not vary from one part of the body to the other.
  • The angular momentum L and the angular velocity ω are not necessarily parallel vectors. However, for the simpler situations discussed in this chapter when rotation is about a fixed axis which is an axis of symmetry of the rigid body, the relation L = Iω holds good, where I is the moment of the inertia of the body about the rotation axis.




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