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System of Particles
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Page 1 Constrains Constrains: Condition or restrictions imposed on motion of particle/particles Motion of particle not always remains free but often is subjected to given conditions. A particle is bound to move along the circumference of an ellipse in XZ plane. At all position of the particle, it is bound to obey the condition + =1 X Z Page 2 Constrains Constrains: Condition or restrictions imposed on motion of particle/particles Motion of particle not always remains free but often is subjected to given conditions. A particle is bound to move along the circumference of an ellipse in XZ plane. At all position of the particle, it is bound to obey the condition + =1 X Z Classification of constrains ? Non-holonomic constrains : Constrains which are not holonomic Two types of constrains are there in this category (i) Equations involving velocities: , … , , … . , , = , (& those cannot be reduced to the holonomic form!). (ii) Constraints as in-equalities, An example, , … . , , < ? Holonomic Constrains: Expressible in terms of equation involving coordinates and time (may or may not present), I,e. , … . , = ; where are the instantaneous coordinates In both type of constrains (holonomic/non-holonomic) time may or may not be present explicitly. Page 3 Constrains Constrains: Condition or restrictions imposed on motion of particle/particles Motion of particle not always remains free but often is subjected to given conditions. A particle is bound to move along the circumference of an ellipse in XZ plane. At all position of the particle, it is bound to obey the condition + =1 X Z Classification of constrains ? Non-holonomic constrains : Constrains which are not holonomic Two types of constrains are there in this category (i) Equations involving velocities: , … , , … . , , = , (& those cannot be reduced to the holonomic form!). (ii) Constraints as in-equalities, An example, , … . , , < ? Holonomic Constrains: Expressible in terms of equation involving coordinates and time (may or may not present), I,e. , … . , = ; where are the instantaneous coordinates In both type of constrains (holonomic/non-holonomic) time may or may not be present explicitly. Pendulum Y X ? s + " = # = # - " % ? One can not change independently, any change in will automatically change ". &, ' ()* + independent due to presence of constrains ( , ") Independent coordinates: If you fix all but one coordinate and still have a continuous range of movement in the free coordinate. If you fix " . , leaving . free, then there is no continuous range of . possible. In fact in this case there will not be any motion if you fix " . Page 4 Constrains Constrains: Condition or restrictions imposed on motion of particle/particles Motion of particle not always remains free but often is subjected to given conditions. A particle is bound to move along the circumference of an ellipse in XZ plane. At all position of the particle, it is bound to obey the condition + =1 X Z Classification of constrains ? Non-holonomic constrains : Constrains which are not holonomic Two types of constrains are there in this category (i) Equations involving velocities: , … , , … . , , = , (& those cannot be reduced to the holonomic form!). (ii) Constraints as in-equalities, An example, , … . , , < ? Holonomic Constrains: Expressible in terms of equation involving coordinates and time (may or may not present), I,e. , … . , = ; where are the instantaneous coordinates In both type of constrains (holonomic/non-holonomic) time may or may not be present explicitly. Pendulum Y X ? s + " = # = # - " % ? One can not change independently, any change in will automatically change ". &, ' ()* + independent due to presence of constrains ( , ") Independent coordinates: If you fix all but one coordinate and still have a continuous range of movement in the free coordinate. If you fix " . , leaving . free, then there is no continuous range of . possible. In fact in this case there will not be any motion if you fix " . Degree of Freedom &Generalized coordinate Y X / ? If you choose / as the only coordinate, it can represent entire motion of the bob in XY plane ? In this problem, only one coordinate / is sufficient which is sole independent coordinate. Degree of Freedom (DOF): no of independent coordinate required to represent the entire motion = 3 × 2 3 4# - . 2 4 =3-2=1 In this case no. of particle=1 No. of constrains =2 [ + " = # and 5 = 0] DOF =1; Generalized Coordinate= / Page 5 Constrains Constrains: Condition or restrictions imposed on motion of particle/particles Motion of particle not always remains free but often is subjected to given conditions. A particle is bound to move along the circumference of an ellipse in XZ plane. At all position of the particle, it is bound to obey the condition + =1 X Z Classification of constrains ? Non-holonomic constrains : Constrains which are not holonomic Two types of constrains are there in this category (i) Equations involving velocities: , … , , … . , , = , (& those cannot be reduced to the holonomic form!). (ii) Constraints as in-equalities, An example, , … . , , < ? Holonomic Constrains: Expressible in terms of equation involving coordinates and time (may or may not present), I,e. , … . , = ; where are the instantaneous coordinates In both type of constrains (holonomic/non-holonomic) time may or may not be present explicitly. Pendulum Y X ? s + " = # = # - " % ? One can not change independently, any change in will automatically change ". &, ' ()* + independent due to presence of constrains ( , ") Independent coordinates: If you fix all but one coordinate and still have a continuous range of movement in the free coordinate. If you fix " . , leaving . free, then there is no continuous range of . possible. In fact in this case there will not be any motion if you fix " . Degree of Freedom &Generalized coordinate Y X / ? If you choose / as the only coordinate, it can represent entire motion of the bob in XY plane ? In this problem, only one coordinate / is sufficient which is sole independent coordinate. Degree of Freedom (DOF): no of independent coordinate required to represent the entire motion = 3 × 2 3 4# - . 2 4 =3-2=1 In this case no. of particle=1 No. of constrains =2 [ + " = # and 5 = 0] DOF =1; Generalized Coordinate= / Degree’s of freedom ? Degree’s of freedom (DOF): No. of independent coordinates required to completely specify the dynamics of particles/system of particles is known as degree’s of freedom. ?Degree’s of freedom = 3 × . 2 3 4# - 7. 2 8#9 4 4 = 37 - : Where 7= No. of particles : =No. of constrains.Read More
FAQs on System of Particles
| 1. What is a system of particles in physics? |  |
Ans. A system of particles refers to a collection of two or more particles that are considered together to analyze their collective behavior and interactions. In physics, this concept is essential for understanding concepts like momentum, energy, and forces acting on the system as a whole.
| 2. How do you calculate the center of mass for a system of particles? | |
Ans. The center of mass (CM) of a system of particles can be calculated using the formula: CM = (m1*r1 + m2*r2 + ... + mn*rn) / (m1 + m2 + ... + mn), where m represents the mass of each particle and r represents its position vector. The CM gives the average position of all the mass in the system.
| 3. What is the significance of the center of mass in a system of particles? | |
Ans. The center of mass is significant because it acts as a point where the total mass of the system can be considered to be concentrated for the purpose of analyzing motion. It simplifies the study of dynamics, as external forces acting on the system will influence its motion primarily through the center of mass.
| 4. Can the center of mass of a system of particles be outside the particles themselves? | |
Ans. Yes, the center of mass of a system of particles can be located outside the physical boundaries of the particles. This occurs in systems where the distribution of mass is uneven, such as in a ring or a hollow shell, where the CM may lie at the center of the shape while the actual mass is distributed along its perimeter.
| 5. How does the conservation of momentum apply to a system of particles? | |
Ans. The conservation of momentum states that in a closed system with no external forces, the total momentum before any event (like a collision) is equal to the total momentum after the event. For a system of particles, this means that the vector sum of the momenta of all particles remains constant, allowing for the analysis of interactions and collisions within the system.
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