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The moment of a force about a point is the turning effect of the force about the point and is measured by the product of of the force and the perpendicular distance of the point from the line of action of the frce. Units : C G S system : Dyne-cm S I system : Newton-meter The moment of a force about a point on its line of action is zero , since the perpendicular distance of the point from the line of action of the force is zero. Couple : A pair of equal , unlike , parallel and noncollinear forces acting on a rigid body is called a couple . Moment of couple : The moment of a couple is equal to the product of one of the force and the perpendicular distance between the lines of action of the forces. The moment of couple is also called Torque. It is a vector. Units : C G S system : Dyne-cm S I system : Newton-m Mometn of Inertia :The moment of inertia of a rigid body about a given axis is the sum of the products of the mass of each particle and the square of its distance from the axis of rotation that make up the body. Let us consider the rotation of a rigid body about fixed axis . If m1,m2,m3 etc. represent the masses of its particles. m1 + m2 + m3 + ………………………. = å m = M where M is the total mass of the body. The moment of inertia of the particle about the axis of rotation at a perpendicular distance r1 from the axis is defined as m1r12 . The moment of inertia of the whole body about the axis is I = m1r12 + m2r22 + m3r32 + ……………………………….. = å mr2 = M å r 2 or I = Mk2 ………………………….(1) Where k is the radius of gyration of the body about the axis of rotation .The radius of gyration is the effective distance of the particles from the axis of rotation. Units of moment of inertia : C G S system : gm . Cm2 S I system : kg . m2 Radius of Gyration (K) : Radius of gyration is the distance of the point from the axis of rotation of a body at which its whole mass M appears to be concentrated so that MK2 becomes equal to the moment of inertia of the body about that axis. Parallel axes theorem : The moment of inertia of a body about an axis is equal to the sum of the of the moments of inertia about a parallel axis passing through its center of gravity , and the product of its mass and the square of the perpendicular distance between the two axes. Or I = Ig + Md2 Where Ig is the moment of inertia of the body about an axis passing through the center of gravity of a body of mass M,I is the moment of inertia about a parallel axis at a distance d from the center of gravity . Perpendicular axes theorem : The sum of the moments of inertia of a plane lamina about any two perpendicular axes in its plane is equal to its moment of inertia about an axis perpendicular to its plane and passing through the point of intersecting of the first two axes or Iz = Ix + Iy. Moment of inertia of bodies with regular geometrical shapes : a) A thin rod of length l and mass M : 1) axis through the center and perpendicular to the rod, I = Ml2 / 12 ………………………….(1) and k2 =l2 / 12 ……………………………(2) 2) Axis through the end and perpendicular to the rod I = Ml2 / 3 and k2 =l2 / 3 b) Circular Ring : A ring of radius R and mass M . About an axis through the center and perpendicular to the plane of the ring. I = MR2 and k2 = R2 c) Circular disc : A disc of mass M and radius R. About an axis passing through its center and perpendicular to its plane face = ½ MR2 ……………………………..(1) and k2 = R2 / 2………………………………(2) d) Cylinder : A solid cylinder of mass M and radius R 1) About an axis passing through the axis of the cylinder. I = MR2 / 2 ………………….(1) and k2 = R2 / 2 ………………….(2) For hollow cylinder I = MR2 2) Solid cylinder about an axis through the center and perpendicular to the axis of symmetry. I = M ( R2 / 4 + l2 / 12 ) and k2 = R2 / 4 + l2 / 1<2 l is the length of the cylinder. e) Solid Sphere :A solid sphere of mass M and radius R about an axis passing through its diameter = 2/5 MR2 ………………….(1) and k2 = 2/5 R2 …………………….(2) f) Hollow Sphere : A hollow sphere of mass M and radius R about an axis passing through its diameter I = 2/3 MR 2 ……………….(1) and k2 = 2/3 R 2 ……………(2) g) Rectangular Lamina : 1) M . I of a rectangular lamina about a line parallel to one of its sides is I = Ml2 / 12 2)The M . I of the plate about an axis parallel to the length and passing through O is I = Mb2 / 12 3)The M . I of the lamina about O perpendicular to its plane is I = Ml2 / 12 + Mb2 / 12 = M ( l2 + b2 / 12 ) 4) The moment of inertia about one edge is I = Ml2 / 3 or I = Mb2 / 3 (similar to thin rod) Angular momentum :The angular momentum or moment of momentum of a body is the product of the moment of inertia (I) and angular velocity ( w ) . This is vector whose direction is same as that of angular velocity ( w ). It is denoted by L. L = I w Units : C G S system : gm cm2 s-1 S I system : Kg m2 s-1 Dimensional Formula : M1L2T-1 Conservation of Angular Momentum : The angular momentum about an axis of a rotating body or system of bodies is constant, if no external couple acts about that axis. Torque on a rotating body :The moment of a force about a point is called the torque and is given by the product of the force( F ) and its perpendicular distance ( r) from the point . Or torque t = F * r Torque is vector quantity . If q is the angle between r and F then t = Fr sin q . If a particle of mass m rotates in a circular path of radius r under the action of a force F , and a is the angular acceleration of the particle , t = F * r = mar Where a is the linear acceleration of the particle. But a = ra , t = F * r = mr2a But mr2 is the moment of inertia I of the particle about the axis of rotation . t = Fr = Ia . K . E of Rotating body :If I is the moment of inertia of a body about its axis of rotating and w is the angular velocity of the body, the kinetic energy of the body is K = ½ Iw2 Work done on a rotating body : W = tq Where W is the work done on the rotating body and q is the angular displacement of the rotating body. The work done on the rotating body is equal to the change in the K.E of the rotating body. Or W = tq = ½ I ( w2 - w02 ) Where I is the moment of inertia of the rotating body about the axis of rotation , w0 is the initial angular velocity and w is the final angular velocity of the body. Relation between angular momentum and K.E of a rotating body : E = ½ L2 / I Relation between angular momentum and Torque : t = dL / dt |