physics_rotational_motion
 Rotational Motion and System of particles Rigid BodyRigid body is a body with a perfectly definite and unchanging shape. The distances between different pairs of such a body do not change. Translational motion In pure translational motion, at any instant of time every particle of the body has the same velocity. Rotational MotionEvery particle of the body moves in a circle in a plane about a fixed axis. The fixed axis and the plane in which body moves are perpendicular. The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or acombination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is rotation. In pure translation of a body, all parts of the body having the same velocity at any instant of time. In pure rotation , all parts of the body having the same angular velocity at any instant of time. The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or acombination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is rotation. (The rotation may be about an axis that is fixed. Or axis may not be fixed but one point may be fixed) Spin Spin is rotation about an axis that goes through the center of mass of the object. Characteristics of Circular Motion1) It is an accelerated motion: As the direction of velocity changes at every instant, it is an accelerated motion.2) It is a periodic motion: During the motion, the particle repeats its path along the same trajectory. Thus, the motion is periodic. Kinematics of Circular Motion1. Angular displacement2. Angular velocity3. Angular acceleration4. Tangential velocity5. Uniform Crcular Motion6. Non Uniform Circular Motion Angular VelocityAngular velocity mesures how fast an object rotates or revolves relative to a given point.How the angle of a vector changes with time with respect to an origin. Spin angular velocity means how fast a rigid body rotates with respect to its centre of rotation. Orbital angular tells us how fast a point object revolves about a given origin. Spin angular velocity is independent of the choice of origin.Orbital angular velocity which depends on the choice of origin. Dynamics of Circular Motion1. Centripetal Force 2. Centrifugal Force Applications of Uniform Circular MotionVehicle Along a Horizontal Circularvmax=𝜇s r g ${v}_{max}=\sqrt{{𝜇}_{s}\phantom{\rule{0.22em}{0ex}}r\phantom{\rule{0.22em}{0ex}}g\phantom{\rule{0.22em}{0ex}}}$ Applications of Uniform Circular Motion1. Well of deathvmin=r g𝜇s ${v}_{min}=\sqrt{\frac{r\phantom{\rule{0.22em}{0ex}}g}{{𝜇}_{s}}\phantom{\rule{0.22em}{0ex}}}$ Applications of Uniform Circular Motion1. . Vehicle on a Banked Road (ignoring friction)Safe Speed = vs=rgtan𝜃${v}_{s}=\sqrt{rgtan𝜃}$Banking Angle, 𝜃=tan-1v2rg$𝜃=ta{n}^{-1}\frac{{v}^{2}}{rg}$ Applications of Uniform Circular Motion1. Vehicle on a Banked Road (consider friction)Minimum Speed = vmin=rg(tan𝜃-𝜇s1+𝜇stan𝜃)${v}_{min}=\sqrt{rg\left(\frac{tan𝜃-{𝜇}_{s}}{1+{𝜇}_{s}tan𝜃}\right)}$ Applications of Uniform Circular Motion1. Vehicle on a Banked Road (consider friction)Maximum Speed = vmax=rg(tan𝜃+𝜇s1-𝜇stan𝜃)${v}_{max}=\sqrt{rg\left(\frac{tan𝜃+{𝜇}_{s}}{1-{𝜇}_{s}tan𝜃}\right)}$ Applications of Uniform Circular MotionConical PendulumTime Period, T=2𝜋L cos𝜃g$T=2𝜋\sqrt{\frac{L\phantom{\rule{0.22em}{0ex}}cos𝜃}{g}}$ Point Mass Undergoing Vertical Circular Motion Under Gravity Vehicle at the Top of a Convex Over Bridge Moment of Inertia as an Analogous Quantity for Mass Moment of InertiaAnalogue of mass in rotational motion.I=n∑i=1mir2i$I=\sum _{i=1}^{n}{m}_{i}{r}_{i}^{2}$ Moment of inertia idepends on the orientation and position of the axis of rotation with respect to the body. Theorem of perpendicular axisMoment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.IZ=IX+IY${I}_{Z}={I}_{X}+{I}_{Y}$ Theorem of parallel axisThe moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes. IZ'=IZ+Ma2${I}_{Z}\text{'}={I}_{Z}+M{a}^{2}$ Angular MomentumRotational analogue of linear momentumIt is moment of linear momentuml=r×p $\stackrel{\to }{l}=\stackrel{\to }{r}×\stackrel{\to }{p}\phantom{\rule{0.22em}{0ex}}$l=I 𝜔 $\stackrel{\to }{l}=I\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{𝜔}\phantom{\rule{0.22em}{0ex}}$dldt=𝜏 $\frac{d\stackrel{\to }{l}}{dt}=\stackrel{\to }{𝜏}\phantom{\rule{0.22em}{0ex}}$ Angular momentum for a system of particlesL=l1+l2+l3+ . . . .+ln =n∑i=1li$\stackrel{\to }{L}=\stackrel{\to }{{l}_{1}}+\stackrel{\to }{{l}_{2}}+\stackrel{\to }{{l}_{3}}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.+\stackrel{\to }{{l}_{n}}\phantom{\rule{0.44em}{0ex}}=\sum _{i=1}^{n}{l}_{i}$L=n∑i=1ri×pi $\stackrel{\to }{L}=\sum _{i=1}^{n}\stackrel{\to }{{r}_{i}}×\stackrel{\to }{{p}_{i}}\phantom{\rule{0.22em}{0ex}}$ Torque1. Rotational analogue of force2. Moment of force3. Torque is the rate of change of angular momentum4. Torque of a force is a measure of that force's tendency to cause a angular acceleration. 𝜏=r×F$\stackrel{\to }{𝜏}=\stackrel{\to }{r}×\stackrel{\to }{F}$𝜏=I 𝛼$\stackrel{\to }{𝜏}=I\phantom{\rule{0.22em}{0ex}}\stackrel{\to }{𝛼}$ Torque for a system of particles L=n∑i=1li $\stackrel{\to }{L}=\sum _{i=1}^{n}{l}_{i}\phantom{\rule{0.22em}{0ex}}$ dLdt=n∑i=1dlidt =n∑i=1𝜏i $\frac{d\stackrel{\to }{L}}{dt}=\sum _{i=1}^{n}\frac{d{l}_{i}}{dt}\phantom{\rule{0.22em}{0ex}}=\sum _{i=1}^{n}{𝜏}_{i}\phantom{\rule{0.22em}{0ex}}$dLdt=𝜏ext $\frac{d\stackrel{\to }{L}}{dt}={𝜏}_{ext}\phantom{\rule{0.22em}{0ex}}$ Conservation of angular momentumif no external torque acts then 𝜏ext=0${𝜏}_{ext}=0$dLdt=𝜏ext=0 $\frac{d\stackrel{\to }{L}}{dt}={𝜏}_{ext}=0\phantom{\rule{0.22em}{0ex}}$ soL=constant$\stackrel{\to }{L}=constant$ Rolling Motion(i) circular motion of the body as a whole, about its own symmetric axis and(ii) linear motion of the body assuming it to be concentrated at its centre of mass. The centre of mass performs purely translational motion. Rolling Motioni) Kinetic energy of a rolling body can be separated into kinetic energy of translation and kineticenergy of rotation. Centre of massCentre of mass is a hypothetical point at which entire mass of the body can be assumed to be concentrated. Centre of mass of a system of particles moves as if all the mass of the system was concentrated at the centre of mass and all the external forces were applied at that point. Centre of mass is a fixed property for given rigid body in spite of any orientation.The centre of gravity may depend upon non-uniformity of the gravitational field, in turn, will depend upon the orientation. Total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its centre of mass. Uniform circular motion : object travels in a circle with a constant speed Centre of gravityCentre of gravity of a body is the point around which the resultant torque due to force of gravity on the body is zero. The turning effect of a force is called the moment of the force. It depends on both the size of the force and how far it is applied from the pivot or fulcrum. Equilibrium of rigid bodyF1+F2+F3+ . . . .+Fn =n∑i=1Fi=0 $\stackrel{\to }{{F}_{1}}+\stackrel{\to }{{F}_{2}}+\stackrel{\to }{{F}_{3}}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.+\stackrel{\to }{{F}_{n}}\phantom{\rule{0.44em}{0ex}}=\sum _{i=1}^{n}{F}_{i}=0\phantom{\rule{0.22em}{0ex}}$𝜏1+𝜏2+𝜏3+ . . . .+𝜏n =n∑i=1𝜏i=0 $\stackrel{\to }{{𝜏}_{1}}+\stackrel{\to }{{𝜏}_{2}}+\stackrel{\to }{{𝜏}_{3}}+\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}.+\stackrel{\to }{{𝜏}_{n}}\phantom{\rule{0.44em}{0ex}}=\sum _{i=1}^{n}{𝜏}_{i}=0\phantom{\rule{0.22em}{0ex}}$ CoupleA pair of equal and opposite forces with different lines of action is known as a couple. A couple produces rotation without translation. The moment of a couple does not depend on the point about which you take the moments. Kinematics of rotational motion Dynamics of rotational motionFor calculation of torque in rigid bodyi) consider only those forces that lie in the plane perpenducular to the axis.ii) forces that are parallel to the axis will give torque perpendicular to the axis.