Tiêu đề ROTATIONAL MOTION.pdf ✅

7 ROTATIONAL MOTION 08 ROTATIONAL MOTION
ROTATIONAL MONTION 1 Chapter 08 Rotational Motion 1. Kinematic of the System of Particles System of particles can move in different ways as observed by us in daily life. To understand this, we need to understand few new parameters. Rigid body: A body in which distance between any two particles remain same regardless of any external changes. 1.1 Kinematic of Rotational Motion (i) Angular Displacement Consider a particle moving from A to B in the following figures. Fig. 8.1 Angle  is the angular displacement of the particle about O. Unit: radian (rad). (ii) Angular Velocity The rate of change of angular displacement is called as angular velocity. Fig. 8.2 Instantaneous Angular Velocity d dt  = Average Angular Velocity t  =  Unit → Rad/s. Angular velocity is a vector quantity whose direction is given by right hand thumb rule. According to right hand thumb rule, if we curl the fingers of right hand along the direction of angular displacement then the right-hand thumb gives us the direction of angular velocity. It is always along the axis of the rotation. (iii) Angular Acceleration Angular acceleration of an object about any point is rate of change of angular velocity about that point. Fig.8.3 2 2 d d dt dt    = = d d d . dt d d     = =    avg t   =  Unit → Rad/s2 . Angular acceleration is also a vector quantity. If  is constant, then like equations of translatory motion we can also write relations between    and t. 0 2 2 0 0 1 2 t t 2 2 t     = +  =  +   −  =  Here, 0 is initial angular velocity and  is final angular velocity. 1.2 Various Types of Motion (i) Translational Motion A system is said to be in translational motion, if all the particles within the system have same linear velocity SCAN CODE Rotational Motion
ROTATIONAL MOTION 139 Example: Motion of a rod as shown below. Fig. 8.4 Example: Motion of body of car on a straight rod. Fig.8.5 In both the above examples, velocity of all the particles is same as they all have equal displacements in equal intervals of time. (ii) Rotational Motion An object is said to be in pure rotational motion, when all the points lying on the system are in circular motion about one common fixed axis. Fig.8.6 In pure rotational motion, angular velocity of all the points is same about the fixed axis. (iii) Rotational + Translational motion An object is said to be in rotational + translational motion, when the particle is rotating with some angular velocity about a movable axis. For Example Fig.8.7 v = velocity of axis.  = Angular velocity of system about O. 1.3 Relationship Between Kinematics Variables In general, if a body is rotating about any axis (fixed or movable), with angular velocity  and angular acceleration , then velocity of any point p with respect to axis is p t r t r v r a a a a r a v =  = + =  =  Fig.8.8 Example Fig.8.9 B A L v L and v , 2  =  = with directions as shown in the figure above. Now in rotational + translational motion, we just superimpose velocity and acceleration of axis on the velocity and acceleration of any point about the axis of rotation. (i.e.) Fig.8.10 PO O v Ri v vi =  = ( ) P O PO P PO O v v v v v v R v i − =  = + =  + SCAN CODE Rotational Motion
Similarly, v R j QO =  O Q v vi v vi R j =  = +  2. Rotational Dynamics 2.1 Torque Similar to force, the cause of rotational motion is a physical quantity called a torque/moment of force/angular force. Torque incorporates the following factors. • Amount of force. • Point of application of force. • Direction of application of force. Combining all the above, Torque about point O, r F r.Fsin  =   =  Where, r = distance from the point O to point of application of force. F = force  = angle between r and F Fig. 8.11 Magnitude of torque can also be rewritten as rF or r F ⊥ ⊥  =  = Where, F⊥ = component of force in the direction perpendicular to r. r⊥ = component of distance in the direction perpendicular to F. (i) Direction of Torque: Direction of torque is given by right hand thumb rule. If we curl the fingers of right hand from first vector ( )r to the second vector ( ) F then right- hand thumb gives us direction of their cross product, i.e., the torque. (ii) Some Important Points about Torque: Torque is always defined about a point or about an axis. When there are multiple forces, the net torque needs to be calculated. i.e., all torque about same point/axis. 1 2 n net F F F  =  +  +  ... • If  = 0 , then the body is said to be in rotational equilibrium. • If  = F 0 along with  = 0 , then body is said to be in mechanical equilibrium (Translation and rotational equilibrium). • If two forces of equal magnitude, opposite direction and do not share a line of action act to produce same torque, then they constitute a couple. It does not produce any translation, only rotation. • For calculating torque, it is very important to know the effective point of application of force. 2.2 Newton's Law in Rotation  = I Where, I = moment of Inertia  = Angular Acceleration 3. Moment of Inertia Moment of inertia gives the measure of mass distribution about an axis. 2 i i I m r =  Where i r = Perpendicular distance of the th i mass from the axis of rotation. Moment of inertia is always defined about an axis. Fig.8.12 For example, moment of inertia for above case, 2 2 2 2 1 1 2 2 3 3 4 4 I M r M r M r M r = + + + • SI unit → kg-m2 • Gives the measure of rotational inertia and is analogous to mass in linear motion. SCAN CODE Rotational Motion 140 ROTATIONAL MOTION