XI PHYSICS. M. Affan Khan LECTURER PHYSICS, AKHSS, K.

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XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@ive.

discussed Rotationa Kinematics i.e. Anguar Dispacement, Veocity, Acceeration etc.

1 XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@ive.com [TORQUE, ANGULAR MOMENTUM & EQUILIBRIUM] CHAPTER NO. 5 Okay here we are going to discuss Rotationa Dynamics actuay, so far we have discussed Rotationa Kinematics i.e. Anguar Dispacement, Veocity, Acceeration etc. and now we wi be continuing further in dynamics. One of the most important and difficut concept we are going to earn is direction of Anguar Momentum.

2 Centre of Mass Centre of mass of a body or a system of partices is defined to be a point where compete mass of the body is supposed to be concentrated such that a the appied forces act at that point. This means that we can describe the motion of the whoe system or the body by the motion of their center of mass. y m2 y2 yc m1 y1 x1 xc x2 x Expanation: Consider this dumbbe shaped object as in figure, having masses m 1 and m 2 on the both sides of the rod. Let the object be acted upon by a number of forces. In order to describe the motion of this object as a whoe we assume that these forces were acting at the center of mass which is the geometrica center of the object and where the tota mass is supposed to be concentrated. We then, find the resutant of these forces and appy Newton s second aw of motion to determine the acceeration and hence the veocity of the center of mass at any instant of time by using initia condition of motion. The motion of the object is same as the motion of the center of mass. Let x & y be the coordinates of the center of mass of that object then they are given by, x x m x m m x m x m x m x m m m m And,

3 y y m y m m y m y m y m y m m m m Simiary, if we have 3-D object then there wi aso be a co-ordinate of z-axis that wi be, z z m z m m z m z m z m z m m m m Moment of Inertia Moment of Inertia of a body is its inabiity to change its condition of rotationa motion. Law of Moment of Inertia: mr A body at rest remains at rest or a body in rotationa motion keeps rotating with uniform anguar veocity uness and unti it is acted upon by some externa torque.

2) We can aso increase that perpendicuar distance with the hep of some spanner extender for this job, which wi aso hep in oosening of that nut as in (b).

4 Torque Torque is defined as, The turning effect of a force. Expanation: Consider a spanner used to tighten or oosen a nut as given in figure. We be providing some turning force to spanner, so we can perform this task in two ways. 1) We can increase the force appied to oosen it easiy as shown in figure (a). 2) We can aso increase that perpendicuar distance with the hep of some spanner extender for this job, which wi aso hep in oosening of that nut as in (b). 100 N 200 N (b) So we may concude that the turning force i.e. Torque depends upon two factors, one is appied force & other is perpendicuar distance aso caed as Moment Arm. Torque orce ppied oment rm r Torque is aso defined as the Vector product of moment arm and force appied, therefore it s a vector quantity and its direction wi be perpendicuar to the pane of both vectors according to right hand rue. t is denoted by Tau ) Its S.I unit is N. m Its dimensions are T

5 Coupe When two forces acting on a body which are equa in magnitude and opposite in direction and acting aong different ine of action such that they have some perpendicuar distance between them are caed as Coupe of orces or simpe Coupe. Consider an object AB aigned at an ange with respect to horizonta. Let us say the position of point A from origin is r and the position of point B from origin is r. And et the ength of object be r. Two forces that are equa in magnitude but opposite in direction are appied at points A & B such that the body start rotating aong its center of mass. The torque at point A wi be r And the torque at point B wi be r Or, r The tota torque acting on the body wi be, (r ) ( r ) r r r r According to diagram, vectors can be added by head to tai rue r r r Or r r r Therefore equation becomes

r r sin r sin r sin Here r sin is perpendicuar distance between the ine of action of forces & as shown in diagram. Let us denote this distance by d d r sin.

forming coupe oment arm of the coupe Since r is the dispacement vector from to, it is independent of the ocation of origin.

Equiibrium: Equiibrium has two types: 1) Static Equiibrium 2) Dynamic Equiibrium Static Equiibrium A body is said to be in equiibrium if it is at rest or is moving with uniform veocity.

6 r r sin r sin r sin Here r sin is perpendicuar distance between the ine of action of forces & as shown in diagram. Let us denote this distance by d d r sin. Then the above equation can be written as, d Where the perpendicuar distance is caed the moment arm of the coupe therefore, agnitude of the omentu of a Coupe agnitude of any of the forces forming coupe oment arm of the coupe Since r is the dispacement vector from to, it is independent of the ocation of origin. Hence the moment of a given coupe is independent of the ocation of origin. Equiibrium: Equiibrium has two types: 1) Static Equiibrium 2) Dynamic Equiibrium Static Equiibrium A body is said to be in equiibrium if it is at rest or is moving with uniform veocity. A body is said to be in static equiibrium is at rest. E.g. this bock in diagram (1) is in static equiibrium. Dynamic Equiibrium (1) Static Equiibrium When body is moving with uniform veocity, it is said to be in dynamic equiibrium. E.g. In figure (2), the person is trying to go up eevator whie eevator is coming downwards so person is moving with constant veocity but he is no doing any work. (2) Dynamic Equiibrium

7 States of Equiibrium: 1) Stabe Equiibrium: When a body doesn t change its position after a sight disturbance, then the body is said to be in stabe equiibrium. 2) Unstabe Equiibrium: When a body oses is origina position after a sight disturbance, then the body is said to be in unstabe equiibrium. 3) Neutra Equiibrium: When a body acquires new position simiar to its origina position, then the body is said to be in neutra equiibrium. Conditions of Equiibrium: 1 st Condition: Mathematicay, When sum of a forces acting on a body is equa to zero then the body is said to be in transationa equiibrium. Proof:,, Let,,, be the n externa forces acting on a body. The first condition of equiibrium states that Force vectors can be written in component form, i j k i j k i j k Adding simiar components ( )i ( )y ( )z

8 Comparing components i, j, k from both sides, i-components j-components k-components In summation form, i-components OR j-components OR k-components OR 2 nd Condition: Mathematicay, When the sum of cockwise and anti-cockwise torques acting on a body is zero then the body wi be in rotationa equiibrium. Since we know that net torque acting on a body produces rotationa motion. Let,,, acting on a body then if, Then the body is said to be in rotationa equiibrium.

9 Anguar Momentum: It is denoted by or It is a vector quantity Its S.I. unit is J.s Expanation: The quantity of rotationa motion contained in a body is caed anguar momentum. OR Anguar momentum can aso be defined as the cross product of position vector r and inear momentum P. r P Consider an object possesses inear motion, at some time t it is ocated at position r from origin and with inear veocity v, so it wi possess inear momentum P. Now if it is subjected to rotate for a distance so it wi have to change its direction, now we can resove position vector into two components, one which is responsibe for its inear motion i. e. r cos which is not required in this case and other one which is responsibe for rotationa motion i. e. r sin as given in figure. As it starts rotating it wi acquire a quantity caed as anguar momentum. So mathematicay we can say that when we mutipy r sin with inear momentum P then object starts rotating. r sin P rp sin Above formua defines when the force is appied at certain ange but if force is appied at 90 0 then, rpsin rp OR rmv If x, y, z are the components of r xi yj zk and P, P, P are the components of P then using definition of vector product we write r P

10 i j k x y z P P P Dimensions of Anguar Momentum, [] [] [r][p] [r][m][v] Unit of Anguar momentum, [ ][ ][ T ] [ T ] mvr kg ( m s ) m kg. m. m s kg. m. m s s s ( kgm s ) m s N m s s Thus, the units of anguar momentum in the S.I. system are Joue-second (Js). Conservation of Anguar Momentum of a Partice: In order to derive the conservation aw for anguar momentum, we obtain a reation between torque and anguar momentum. ccording to Newton s second aw of motion, the net force acting on a partice of mass m moving with an instantaneous veocity v is the time rate of change of its inear momentum P. P t In differentia form,

11 dp Taking vector product with respect to position vector r on both sides, r r dp r dp By definition of anguar momentum, r P Differentiating both sides with respect to time, d d (r P ) dr P r dp v mv m v v d The above reation states that the torque acting on a partice is the time rate of change of anguar momentum. If net externa torque acting on the partice is zero (i.e. condition of rotationa equiibrium), then above equation reduces to, d OR we may say that there is no change in anguar momentum That is constant

12 Appications of Conservation of Anguar Momentum: i) You might have seen a skater undergoing a spin motion in the finae of an act. We assume that there is no friction between the skater and the ice and hence there is no externa torques acting on the skater. Thus the anguar momentum of the skater which is the product of moment of inertia, I, of the skater and the anguar veocity,, of the skater, is constant, since moment of inertia depends on the distribution of mass, the skater can decrease his or her moment of inertia by puing his or her hands and feet cose to the body. As a resut the anguar veocity of the skater increases and so the spinning takes pace at a rapid rate. ii) When divers and acrobats who wish to make severa somersauts, they pu their hands and feet cose to their bodies in order to rotate at a higher rate. Due to the cose distribution of mass the moment of inertia decreases. This causes an increase in the anguar veocity enabing them to make severa somersauts.

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I 6 n terms of moment of inertia, equation (7.8) can be written as The vector form of the above equation is...(7.9 a)...(7.9 b) The anguar acceeration produced is aong the direction of appied externa torque.