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Page 1 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 Page 2 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) Page 3 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? Page 4 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? ?? ???? ???? =?? -?? ?? 2 or ?? ???? ???? +?? ?? 2 =?? Integrating and using initial conditions, we get ?? 2 = ?? ?? [1-?? -2???? ] … (1) It follows that ?? < v ?? ?? and ?? ? v?? ?? as ?? ?8 Integrating ???? ???? =?? -?? ?? 2 and using initial conditions we get ?? =v ?? ?? tanh (?? v???? ) Therefore Page 5 Edurev123 DYNAMICS OF A PARTICLE The subject of Dynamics consists of Kinematics (geometry of motion apart from all considerations of force, mass or energy) and Kinetics (effects of forces on motion of bodies). 1. Rectilinear motion (Kinematics and ?? ?? netics) Let the displacement of moving point ?? measured from ?? be ?? Then the velocity v of P at the instant is ???? ???? and it is positive in the direction of ?? increasing. The acceleration of ?? at that instant is ?? 2 ?? ?? ?? 2 or ?? ???? ???? and it is positive in the direction of ?? increasing. The simplest case is that of a particle moving with uniform acceleration ?? , the displacement and velocity initially (at time ?? =0 ) being zero and u respectively. Then ???? ???? =?? ???? ?? x =?? : Integrating and using initial conditions, one obtains ?? =?? +???? ?? =???? + 1 2 ?? ?? 2 and ?? 2 =?? 2 +2???? We shall illustrate in the following examples, motion when acceleration s not uniform. Example 1: A particle ?? moves in a straight line, its acceleration directed towards a fixed point ?? in the line is ?? ( ?? 5 ?? 2 ) 1 3 , when its displacement from ?? is ?? . If it starts from rest at ?? = ?? , show that it will arrive at ?? at the end of time 8 15 v 6 ?? with velocity a v6?? ?? ???? ???? =-?? ( ?? 5 ?? 2 ) 1 3 the negative sign on the right is due to the fact that the acceleration is towards ?? , hence negative when ?? is positive. Integrating and using initial conditions, when ?? =0,?? =?? , and ?? =0 we get ?? 2 =6?? 5/3 (?? 1/3 -?? 1/3 ) Therefore ?? =-v6?? ?? 5/3 v?? 1/3 -?? 1/3 At ?? =0,?? =-?? v6?? Integrating v6?? ?? 5/3 ?? =-? ?? =?? 0 ? ???? ?? 1/3 -?? 1/3 Therefore ?? = 8 15 v 6 ?? Example 2: A particle moves along a straight-line with an acceleration directed toward fixed ?? point O on it and inversely proportional to the square of the distance of the particle from O. If the particle is at rest initially at a distance 2a from ?? ,?? how that the time of motion from the distance 2?? to the distance ?? is to the time from distince a to O in the ratio ?? +2:?? -2. ?? ???? ???? =- ?? ?? 2 Integrating and using initial conditions. ?? 2 =?? ( 2 ?? - 1 ?? ) or ???? ???? =-v ?? ?? v 2?? -?? ?? Therefore v ?? ?? ?? =-1v ?? 2?? -?? ???? ? -?? v2???? -?? 2 ???? or v ?? ?? ?? =v?? 2 -(?? -?? ) 2 +?? cos -1 ( ?? -?? ?? ) The constant of integration is zero as ?? =0 when ?? =2?? . Time ?? 1 to go from ?? =2?? to ?? = ?? is given by v ?? ?? ?? 1 =?? [1+ ?? 2 ] Time ?? 2 , to go from ?? =?? to ?? =0 is given by v ?? ?? ?? 2 =???? -?? (1+ ?? 2 )=?? ( ?? 2 -1) Therefore ?? 1 ?? 2 = ?? +2 ?? -2 2. Vertical motion under gravity with resistance (a) A particle falls freely from rest under gravity (assumed constant) in a medium whose resistance is proportional to the square of its velocity. Find the distance travelled and or the velocity of the particle at time ?? . When the particle has fallen a distance ?? in time ?? ?? ???? ???? =?? -?? ?? 2 or ?? ???? ???? +?? ?? 2 =?? Integrating and using initial conditions, we get ?? 2 = ?? ?? [1-?? -2???? ] … (1) It follows that ?? < v ?? ?? and ?? ? v?? ?? as ?? ?8 Integrating ???? ???? =?? -?? ?? 2 and using initial conditions we get ?? =v ?? ?? tanh (?? v???? ) Therefore 1-?? -2???? =?? 2 ?? ?? =tanh 2 (?? v???? ) from (1), (2) ?? -2???? =sech 2 (?? v???? ) or ?? ???? =cosh (?? v???? ) or ?? = 1 ?? ln cosh (?? v???? ) (b) A particle is projected vertically upwards with velocity ?? in a medium whose its resistance is proportional to the square of its velocity. To find its motion, assuming gravity to be constant. We have ???? ???? =-?? -?? ?? 2 Integrating and using initial conditions, we get v???? =tan -1 (?? v ?? ?? )-tan -1 (?? v ?? ?? ) Again ?? ???? ???? =-?? -?? ?? 2Read More
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FAQs on Dynamics of a Particle - Mathematics Optional Notes for UPSC
| 1. What is the equation of motion for a particle in dynamics? |  |
Ans. The equation of motion for a particle in dynamics is given by Newton's second law of motion, which states that the sum of all forces acting on a particle is equal to the mass of the particle multiplied by its acceleration.
| 2. How does velocity affect the dynamics of a particle? | |
Ans. Velocity plays a crucial role in the dynamics of a particle as it determines the rate at which the position of the particle changes with time. The velocity of a particle can affect its acceleration and ultimately its motion.
| 3. What is the difference between statics and dynamics in the context of particles? | |
Ans. Statics deals with the study of particles at rest or in equilibrium, where the sum of all forces acting on the particle is zero. On the other hand, dynamics deals with particles in motion and studies the forces that cause the motion.
| 4. How can we calculate the acceleration of a particle in dynamics? | |
Ans. The acceleration of a particle in dynamics can be calculated by dividing the net force acting on the particle by its mass. This is in accordance with Newton's second law of motion, which relates acceleration to force and mass.
| 5. What are the different types of forces that can act on a particle in dynamics? | |
Ans. In dynamics, particles can experience various types of forces such as gravitational force, frictional force, normal force, tension force, and applied force. These forces can influence the motion and behavior of the particle under study.
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