Simple Nature - Light and Matter

angular momentum is related to rate at which area is swept out by the line segment connectingthe particle to the axis.Torque is the rate of change of angular momentum, τ = dL/ dt. The torque created by agiven force can be calculated using any of the relationsτ = rF sin θ rF= rF ⊥= r ⊥ F ,where the subscript ⊥ indicates a component perpendicular to the line connecting the axis tothe point of application of the force.In the special case of a rigid body rotating in a single plane, we defineω = dθdt[angular velocity]andα = dωdt, [angular acceleration]in terms of which we haveL = Iωandτ = Iα ,where the moment of inertia, I, is defined asI = ∑ m i r 2 i ,summing over all the atoms in the object (or using calculus to perform a continuous sum, i.e.an integral). The relationship between the angular quantities and the linear ones isv t = ωrv r = 0[tangential velocity of a point][radial velocity of a point]a t = αr . [radial acceleration of a point]at a distance r from the axis]a r = ω 2 r[radial acceleration of a point]at a distance r from the axis]In three dimensions, torque and angular momentum are vectors, and are expressed in terms ofthe vector cross product, which is the only rotationally invariant way of defining a multiplicationof two vectors that produces a third vector:L = r × pτ = r × F951