Title 7.System of Particles and Rotational Motion-f.pdf ✅

1 | P a g e NEET-2022 Ultimate Crash Course PHYSICS System of Particles and Rotational Motion
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3 | P a g e POINTS TO REMEMBER 1. Centre of mass of a system of particles (or a body) is a point where the whole mass of the system (or a body) is supposed to be concentrated for describing its translatory motion. 2. The position of the centre of mass depends upon the shape of the body. 3. It may be within (as in case of a sphere) or outside the body (as in the case of a ring). 4. There may or may not be any mass actually present at the centre of mass, e.g., the centre of mass of a ring, which is its centre, has no mass there. 5. For a given shape, the centre a mass depends upon the distribution of mass within the body. It lies closer to the massive part (i.e., two spheres when joined by a rod, have their centre of mass closer to the heavier of the two spheres). 6. The centre of mass (CM) and centre of gravity (CG) arc two different points. Whereas centre of mass is the point where the whole mass of the body is supposed to be concentrated, centre of gravity is a point where its whole weight is supposed to act. If a body is in a uniform gravitational field, the two points coincide. Since the force of gravity decreases with altitude, the lower portions of the body have more weight than its upper portions and as such the centre of gravity is a bit lower than the centre of mass. Ordinarily, the difference in the positions of two points is so small that it can he neglected. But it is useful to distinguish between CM and CG. In the case of a spherical ball, the CM and CG are the same, but in the case of Mount Everest, its CM lies a bit above its CG. 7. In a two-particle system, the centre of mass divides the distance between them in the inverse ratio of their masses. 8. If there are two particles of masses m1 and m2, then  = + m m / m m 1 2 1 2 ( ) is known as the reduced mass of the two-particle system. 9. The relative motion of two particles subject only to their mutual interaction is equivalent to the motion of a particle of mass equal t the reduced mass under a force equal to their interaction. 10. The motion of the Moon relative to the Earth can be reduced to a single particle problem by using the reduced mass of the Earth-Moon system and a force equal to the attraction of the Earth on the Moon. 11. When we speak of the velocity of a moving body composed of many particles, such m an aeroplane or an automobile, the Earth or the Moon, or even a molecule or a nucleus, we actually refer to its centre of mass velocity, cm v . Further, cm v is sometimes called system velocity. 12. The centre of mass of an isolated system moves with constant velocity relative to any inertial frame. 13. If 0. 0, cm v p = = i.e., if a reference frame is attached to the centre of mass of a system, the total momentum of the system is zero in that reference frame. Such a reference frame is called the center of mass frame of reference or C-frame of reference. Obviously relative to this frame, the centre of mass is at rest. 14. In any given system of particles, the sum of the internal forces is always equal to zero. 15. If 0. 0, F a and v ext cm cm = = = constant. This implies that the velocity of centre of mass is independent of the internal forces acting between the various particles of the system, i.e., the motion of the centre of mass remains unaffected in the absence of any external force. 16. The time rate of change of momentum of a system of particles is equal to the external force applied to the system, i.e., / F dp dt ext = . 17. The change in kinetic energy of a system of particles is equal to the work done on the system by the external and the internal forces 18. When a rigid body mimes about a fixed axis, every pan of the body has the same angular velocity and angular acceleration. However, different pans of the body. in general, have different linear velocities and different linear accelerations. 19. Moment of inertia of a body has different values in different directions and as such it is not a scalar. Further, it is not a vector either as its value about a given axis remains the sonic whether the direction of rotation is clockwise or anticlockwise, i.e., direction of rotation need not be specified. In fact, it is tensor quantity. 20. Rolling motion of a body is a combination of translatory as well as rotatory motion.
4 | P a g e 21. For all rigid bodies of the same shape. rolling down an inclined plat, (i) the time taken to reach the bottom (ii) velocity and (iii) acceleration do not depend upon theire mass. For example, all solid cylinders would experience the tame velocity and acceleration on a given incline. Same is true for all rings, all discs, all spheres, all shells etc. But note that for the bodies of different shapes, even though of the same mass. these quantities (time, velocity, acceleration) differ. For example, a solid sphere rolls down faster than a spherical shell of the same mass and radius. In general, (i) v v v v v sphere disc solid cylinder sheel ring  =   ( ) (ii) a a a a a sphere disc solid cylinder sheel ring  =   ( ) (iii) Since more the velocity and acceleration, lesser the time taken (t )to reach the bottom of the incline, t t t t t sphere disc solid cylinder sheel ring  =   ( ) Thus, it is the distribution of mass and not the mass which determines v, a and t. 22. Rolling motion is possible only if a frictional force is present between the rolling body and the incline to produce a net torque about the centre of mass 23. Despite the presence of friction, there is no loss of mechanical energy in rolling motion since the contact point is at rest relative to the surface at any instant. On the other hand, if a rigid body were to slide, mechanical energy would be lost as the motion progresses. 24. When a body rolls down an incline plane, its velocity is independent of the inclination of the plane but depends upon the height through which the body descends as v gh cm = + 2 / 1(  ) 25. Acceleration and time of descent depend upon the inclination. Greater inclination produces greater acceleration and therefore reduces the time of descent as a g cm = + sin / 1   ( ) 26. More the MI, more K r and the less Kt , which implies lesser linear velocity. 27. As / 1/ K K t r =  For a hoop (ring), 1,  = = K K t r For a spherical shell, 2 3 , 1.5 3 2  = = = K K K t r r For a disc (or a solid cylinder), 1 , 2 2  = = K K t r For a solid sphere, 2 5 , 2.5 5 2  = = = K K K t r r 28. (I) In pure translational motion, every point on the wheel moves with velocity cm v . The wheel does not rotate at all, shown in the figure (1) (II) In case of pure rotational motion, every point on the rolling wheel has same to All points on the edge of the wheel have same linear velocity R . The centre O is at rest, shown in the figure (2) (III) In rolling motion, the wheel possess both the above motions. The combination of figure (1) and figure (2), gives us the actual rolling motion, Consequently, the velocity at the contact point P is (v R cm − ) and the velocity at the top point (T) is (v R cm + ) In case of rolling without slipping, cm v R = and in that case velocity at P is 0 and velocity at T is 2 cm v . 29. The relations L I =  and   = I are applicable only when the rigid body rotation is about a fixed axis through the centre of mass.