Tiêu đề ROTATIONAL MOTION.pdf ✅

 Digital www.allendigital.in [ 91 ] Introduction Rigid body A rigid body is an assemblage of a large number of material particles, which do not change their mutual distances under any circumstance or in other words, the body is not deformed under any circumstance. Actual material bodies are never perfectly rigid and are deformed under the action of external forces. When these deformations are small enough not to be considered during the course of motion, the body is assumed to be a rigid body. Hence, all solid objects such as stone, ball, vehicles etc are considered as rigid bodies while analysing their translational as well as rotational motion. Rotational motion of a rigid body Any kind of motion is identified by change in position or change in orientation or change in both. If a body changes its orientation during its motion it said to be in rotational motion. In the following figures, a rectangular plate is shown moving in the x-y plane. The point C is its centre of mass. In the first case it does not change its orientation, therefore is in pure translation motion. In the second case it changes its orientation during its motion. It is a combination of translational and rotational motion. Rotation i.e. change in orientation is identified by the angle through which a linear dimension or a straight line drawn on the body turns. In the figure this angle is shown by . Types of motions involving rotation Motion of body involving rotation can be classified into following three categories. I. Rotation about a fixed axis. II. Rotation about an axis in translation. III. Rotation about an axis in rotation Rotation about a fixed axis Rotation of ceiling fan, opening and closing of doors and rotation of needles of a wall clock etc. come into this category. When a ceiling fan rotates, the vertical rod supporting it remains stationary and all the particles on the fan move on circular paths. Circular path of a particle P on one of its blades is shown by dotted circle. Centres of circular paths followed by every particle on the central line through the rod. This central line is known y O Pure Translation x t •C •C t+t y O x Combination of translation and rotation t A B • C t+t B C  A •  Original orientation New orientation 03 Rotational Motion
NEET : Physics [ 92 ] www.allendigital.in  Digital as the axis of rotation and is shown by a dashed line. All the particles on the axis of rotation are at rest, therefore the axis is stationary and the fan is in rotation about this fixed axis. A door rotates about a vertical line that passes through its hinges. This vertical line is the axis of rotation. In the figure, the axis of rotation is shown by dashed line. Axis of rotation An imaginary line perpendicular to the plane of circular paths of particles of a rigid body in rotation and containing the centres of all these circular paths is known as axis of rotation. It is not necessary that the axis of rotation should pass through the body. Consider a system shown in the figure, where a block is fixed on a rotating disc. The axis of rotation passes through the center of the disc but not through the block. Important observations Let us consider a rigid body of arbitrary shape rotating about a fixed axis PQ passing through the body. Two of its particles A and B are shown moving on their circular paths. All its particles, not lying on the axis of rotation, move along circular paths with centres on the axis or rotation. All these circular paths are in parallel planes that are perpendicular to the axis of rotation. All the particles of the body undergo same angular displacement in the same time interval, therefore all of them move with the same angular velocity and angular acceleration. Particles moving on circular paths of different radii move with different speeds and different magnitudes of linear acceleration. Furthermore, no two particles in the same plane perpendicular to the axis of rotation have same velocity and acceleration vectors. Rotation about an axis in translation Rotation about an axis in translation includes a broad category of motions. Rolling is an example of this kind of motion. Consider the rolling of wheels of a vehicle, moving on straight levelled road. The wheel appears rotating about its stationary axle relative to a reference frame, attached with the vehicle. The rotation of the wheel as observed from this frame is rotation about a fixed axis. Relative to a reference frame fixed with the ground, the wheel appears rotating about the moving axle, therefore, rolling of a wheel is superposition of two simultaneous but distinct motions – rotation about the axle fixed with the vehicle and translation of the axle together with the vehicle. Ceiling Fan Axis of rotation P Door Axis of rotation Axis of rotation
Rotational Motion  Digital www.allendigital.in [ 93 ] Rotation about an axis in rotation. In this kind of motion, the body rotates about an axis which in turn rotates about some other axis. Analysis of rotation about rotating axes is beyond our scope, therefore we shall keep our discussion elementary level only. As an example consider a rotating top. The top rotates about its central axis of symmetry and this axis sweeps a cone about a vertical axis. The central axis continuously changes its orientation, therefore it is in rotational motion. This type of rotation in which the axis of rotation also rotates and sweeps out a cone is known as precession. Another example of rotation about an axis in rotation is a swinging table-fan while running. Table-fan rotates about its shaft along which its axis of rotation passes. When running swings, its shaft rotates about a certain axis. Note: In a rigid body angular velocity of any point w.r.t. any other point is constant and is equal to the angular velocity of the rigid body. Illustration 1: In Rotational motion of a rigid body, all particles move with (1) same linear and angular velocity (2) same linear and different angular velocity. (3) with different linear velocities & same angular velocities (4) with different linear velocity & different angular velocities. Solution:(3) In rotational motion of the rigid body all particles cover the same angular displacements in a particular interval. So angular velocity of all the particle will be same. But linear velocity is also dependent on the distance of particles from the axis of rotation so linear velocity will be different for all particles as the distances are different for all the particles. Kinematics of Rotational Motion Time period (T) : Time taken by the particle to complete one rotation. frequency (f) : No. of cycles completed by a particle per second is known as frequency. rpm = rotations per minute(N) N f 60 = Angular Displacement () • When a particle moves in a curved path, the change in the angle traced by its position vector about a fixed point is known as angular displacement. • Unit : radian • Dimension : M0L0T0 i.e. dimensionless. • Elementary (small) angular displacement is a vector whereas other (large) angular displacements is a scalar. Angular Velocity () • The angular displacement per unit time is defined as angular velocity. t  =  where  is the angular displacement during the time interval t. • Instantaneous angular velocity t 0 d Lim t dt  →    = =  . Average angular velocity 2 1 av 2 1 t t t  −    = = −  • Unit : rad/s
NEET : Physics [ 94 ] www.allendigital.in  Digital • Dimensions: [M0L0T–1], which is same as that of frequency. • Instantaneous angular velocity is a vector quantity, whose direction is normal to the rotational plane and its direction is given by right hand screw rule. • If be the angular velocity, v the linear velocity and r the radius of path, we have the following relation. v r =  • If n be the frequency then  = 2n, If T be the time period then  = 2/T. • The angular velocity of a rotating rigid body can be either positive or negative, depending on whether it is rotating in the direction of increasing  (anticlockwise) or decreasing  (clockwise). • The magnitude of angular velocity is called the angular speed which is also represented by . Angular Acceleration () • The rate of change of angular velocity is defined as angular acceleration d dt  = • Suppose a particle has angular velocity 1 & 2 at time t1 and t2 respectively then average angular acceleration, 2 1 2 1 t t  −   = − • It is a vector quantity, whose direction is along the change in direction of angular velocity. • Unit : rad/s2 • Dimensions : M0L0T–2 • Relation between angular acceleration  and tangential acceleration t a is t a r =  • Radial or normal acceleration: r a v =  . Its direction is along the radius. • Net acceleration: t r a a a r v = + =  +  Comparison of Linear Motion and Rotational Motion Linear Motion Rotational Motion (i) If acceleration is 0, v=constant & s = vt (i) If angular acceleration is 0,  = constant &  = t (ii) If acceleration a = constant, then (a) (u v) s t 2 + = (b) v u a t − = (c) v = u + at (d) s = ut + 1 2 at2 (e) v2 = u2 + 2as (f) th n S = u + a (2n 1) 2 − (ii) If angular acceleration  = constant, then (a) 0 ( ) t 2  +   = (b) 0 t −   = (c)  = 0 + t (d)  = 0t + 1 2 t 2 (e) 2 = 0 2 + 2 (f) th n  = 0 + (2n 1) 2  − (iii) If acceleration is not constant, the above equation will not be applicable. In this case (a) ds v dt = (b) 2 2 dv d s dv v dt ds dt  = = = (iii) If angular acceleration is not constant, the above equation will not be applicable. In this case (a) d dt  = (b) 2 2 d d d dt d dt     = = =   • In a rigid body, angular velocity of any point w.r.t. any other point is constant and is equal to the angular velocity of the rigid body.