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Page 2 ? A triangular plate (By qualitative argument) at the centroid : y c = 3 h ? A semi-circular ring y c = ? R 2 x c = O ? A semi-circular disc y c = ? 3 R 4 x c = O ? A hemispherical shell y c = 2 R x c = O ? A solid hemisphere y c = 8 R 3 x c = O ? A circular cone (solid) y c = 4 h ? A circular cone (hollow) y c = 3 h Page 3 ? A triangular plate (By qualitative argument) at the centroid : y c = 3 h ? A semi-circular ring y c = ? R 2 x c = O ? A semi-circular disc y c = ? 3 R 4 x c = O ? A hemispherical shell y c = 2 R x c = O ? A solid hemisphere y c = 8 R 3 x c = O ? A circular cone (solid) y c = 4 h ? A circular cone (hollow) y c = 3 h MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM: Velocity of centre of mass of system cm v ? = M dt dr m .... .......... dt dr m dt dr m dt dr m n n 3 3 2 2 1 1 ? ? ? = M v m .......... v m v m v m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? System P = M cm v Acceleration of centre of mass of system cm a ? = M dt dv m .... .......... dt dv m dt dv m dt dv m n n 3 3 2 2 1 1 ? ? ? = M a m .......... a m a m a m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? = M system on force Net = M Force ernal int Net Force External Net ? = M Force External Net ext F ? = M cm a ? IMPULSE Impulse of a force F action on a body is defined as :- J ? = ? f i t t Fdt P ? J ? ? ? (impulse - momentum theorem) Important points : 1. Gravitational force and spring force are always non-impulsive. 2. An impulsive force can only be balanced by another impulsive force. COEFFICIENT OF RESTITUTION (e) e = n deformatio of pulse Im n reformatio of pulse Im = ? ? dt F dt F d r = s impact of line along approach of Velocity impact of line along separation of Velocity Page 4 ? A triangular plate (By qualitative argument) at the centroid : y c = 3 h ? A semi-circular ring y c = ? R 2 x c = O ? A semi-circular disc y c = ? 3 R 4 x c = O ? A hemispherical shell y c = 2 R x c = O ? A solid hemisphere y c = 8 R 3 x c = O ? A circular cone (solid) y c = 4 h ? A circular cone (hollow) y c = 3 h MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM: Velocity of centre of mass of system cm v ? = M dt dr m .... .......... dt dr m dt dr m dt dr m n n 3 3 2 2 1 1 ? ? ? = M v m .......... v m v m v m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? System P = M cm v Acceleration of centre of mass of system cm a ? = M dt dv m .... .......... dt dv m dt dv m dt dv m n n 3 3 2 2 1 1 ? ? ? = M a m .......... a m a m a m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? = M system on force Net = M Force ernal int Net Force External Net ? = M Force External Net ext F ? = M cm a ? IMPULSE Impulse of a force F action on a body is defined as :- J ? = ? f i t t Fdt P ? J ? ? ? (impulse - momentum theorem) Important points : 1. Gravitational force and spring force are always non-impulsive. 2. An impulsive force can only be balanced by another impulsive force. COEFFICIENT OF RESTITUTION (e) e = n deformatio of pulse Im n reformatio of pulse Im = ? ? dt F dt F d r = s impact of line along approach of Velocity impact of line along separation of Velocity (a) e = 1 ? Impulse of Reformation = Impulse of Deformation ? V elocity of separation = V elocity of approach ? Kinetic Energy may be conserved ? Elastic collision. (b) e = 0 ? Impulse of Reformation = 0 ? V elocity of separation = 0 ? Kinetic Energy is not conserved ? Perfectly Inelastic collision. (c) 0 < e < 1 ? Impulse of Reformation < Impulse of Deformation ? V elocity of separation < V elocity of approach ? Kinetic Energy is not conserved ? Inelastic collision. VARIABLE MASS SYSTEM : If a mass is added or ejected from a system, at rate ? kg/s and relative velocity rel v ? (w.r.t. the system), then the force exerted by this mass on the system has magnitude rel v ? ? . Thrust Force ( t F ? ) ? ? ? ? ? ? ? dt dm v F rel t ? ? Rocket propulsion : If gravity is ignored and initial velocity of the rocket u = 0; v = v r ln ? ? ? ? ? ? m m 0 . RIGID BODY DYNAMICS 1. RIGID BODY : A V A V B ? 2 ? 1 B V B sin ? 2 V A cos ? 1 V A sin ? 1 V B cos ? 2 A B Page 5 ? A triangular plate (By qualitative argument) at the centroid : y c = 3 h ? A semi-circular ring y c = ? R 2 x c = O ? A semi-circular disc y c = ? 3 R 4 x c = O ? A hemispherical shell y c = 2 R x c = O ? A solid hemisphere y c = 8 R 3 x c = O ? A circular cone (solid) y c = 4 h ? A circular cone (hollow) y c = 3 h MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM: Velocity of centre of mass of system cm v ? = M dt dr m .... .......... dt dr m dt dr m dt dr m n n 3 3 2 2 1 1 ? ? ? = M v m .......... v m v m v m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? System P = M cm v Acceleration of centre of mass of system cm a ? = M dt dv m .... .......... dt dv m dt dv m dt dv m n n 3 3 2 2 1 1 ? ? ? = M a m .......... a m a m a m n n 3 3 2 2 1 1 ? ? ? ? ? ? ? = M system on force Net = M Force ernal int Net Force External Net ? = M Force External Net ext F ? = M cm a ? IMPULSE Impulse of a force F action on a body is defined as :- J ? = ? f i t t Fdt P ? J ? ? ? (impulse - momentum theorem) Important points : 1. Gravitational force and spring force are always non-impulsive. 2. An impulsive force can only be balanced by another impulsive force. COEFFICIENT OF RESTITUTION (e) e = n deformatio of pulse Im n reformatio of pulse Im = ? ? dt F dt F d r = s impact of line along approach of Velocity impact of line along separation of Velocity (a) e = 1 ? Impulse of Reformation = Impulse of Deformation ? V elocity of separation = V elocity of approach ? Kinetic Energy may be conserved ? Elastic collision. (b) e = 0 ? Impulse of Reformation = 0 ? V elocity of separation = 0 ? Kinetic Energy is not conserved ? Perfectly Inelastic collision. (c) 0 < e < 1 ? Impulse of Reformation < Impulse of Deformation ? V elocity of separation < V elocity of approach ? Kinetic Energy is not conserved ? Inelastic collision. VARIABLE MASS SYSTEM : If a mass is added or ejected from a system, at rate ? kg/s and relative velocity rel v ? (w.r.t. the system), then the force exerted by this mass on the system has magnitude rel v ? ? . Thrust Force ( t F ? ) ? ? ? ? ? ? ? dt dm v F rel t ? ? Rocket propulsion : If gravity is ignored and initial velocity of the rocket u = 0; v = v r ln ? ? ? ? ? ? m m 0 . RIGID BODY DYNAMICS 1. RIGID BODY : A V A V B ? 2 ? 1 B V B sin ? 2 V A cos ? 1 V A sin ? 1 V B cos ? 2 A B If the above body is rigid V A cos ? 1 = V B cos ? 2 V BA = relative velocity of point B with respect to point A. V BA B A Pure Translational Motion Pure Rotational Motion Types of Motion of rigid body Combined Translational and Rotational Motion 2. MOMENT OF INERTIA (I) : Definition : Moment of Inertia is defined as the capability of system to oppose the change produced in the rotational motion of a body. Moment of Inertia is a scalar positive quantity. ? = mr 1 2 + m 2 r 2 2 +......................... = ? ? + ? ? + ? ? +......................... S ? units of Moment of Inertia is Kgm 2 . Moment of Inertia of : 2.1 A single particle : ? = mr 2 where m = mass of the particle r = perpendicular distance of the particle from the axis about which moment of Inertia is to be calculated 2.2 For many particles (system of particles) : ? = ? ? n 1 i 2 i i r m 2.3 For a continuous object : ? = ? 2 dmr where dm = mass of a small element r = perpendicular distance of the particle from the axisRead More
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