Centre Of Mass
Centre of mass is defined as the point at which the total mass of a system is supposed to be concentrated
On the application of external force Centre of mass of system of particles move in the same way as a point having mass equal to that of the whole mass move
Position Of Centre of Mass
1) For Two Particles Rcm = \(\frac{m_1r_1+m_2r_2}{m_1+m_2}\) = \(\frac{m_1r_1 + m_2r_2}{M}\)
2) For Three Dimensional Axis Xcm = \(\frac{m_1x_1 + m_2x_2}{m_1 + m_2}\) = \(\frac{m_1x_1 + m_2x_2}{M}\) = ∑\(\frac{m_ix_i}{ M}\)
\(Y_{cm} =\frac{m_1y_1 + m_2y_2}{m_1 + m_2} = \frac{m_1y_1 + m_2y_2}{M} \)= ∑\(\frac{m_iy_i}{M}\)
Zcm = \(\frac{m_1z_1 + m_2z_2} {m_1 + m_2} = \frac{m_1z_1 + m_2z_2}{M} = ∑\frac{m_iz_i}{M}\)
For Large Number of Particles Rcm = ∑\(\frac{m_ir_i}{M}\)
3)For Continuous Bodies Rcm = \(\frac{∫r^2 dm}{M}\)
In the Coordinates Xcm = \(\frac {∫x^2 dm}{M}\)
Ycm = \(\frac {∫ y^2 dm}{M}\)
Zcm = \(\frac{∫ z^2 dm}{M}\)
Motion of Centre of Mass
1) Vcm = \(\frac{m_1v_1 + m_2v_2 + …………. + m_nv_n}{M}\) = ∑\(\frac{m_iv_i}{M}\)
2) acm = \(\frac{m_1a_1 + m_2a_2 + …………. + m_na_n}{M}\) = ∑\(\frac{m_ia_i}{M}\)
3) Fcm = \(F_1 + F_2 + …………… + F_n\) = ∑ \(F_i\)
If System contains more than one particle and Fext in a particular direction is zero then centre of mass is also zero even though some particles may move along the direction
Motion of center of mass is not affected by internal force because it always makes an action reaction pairs and their contribution to acceleration is zero
If m1 and m2 are placed by separation r and move towards each other by their mutual attraction force then 1) Acm = zero but not individual velocities 2) Vcm = Zero
If larger number of particles are projected under the effect of gravity only in different directions then acceleration of center of mass is equal to the acceleration due to gravity irrespective of directions of projection of particles
The total momentum of a system of particles is equal to product of the total mass and velocity of its centre of mass i.e.. \(P = MV_{cm}\) and we can also define total linear momentum as the sum of linear momenta of individual particles is \(P = P_1 + P_2 + P_3 + ………+ P_n\)
The relation between KE and Momentum is \(\sqrt{2mk}\)
Conservation of linear Momentum for System of particles :
If there is no external force the sum of total linear momentum of system remains constant or conserved
\(P = MV_{cm}\)
By differentiating P and \(M_{Vcm}\) we get \(\frac{dp}{dt}\) = \(F_{ext}\)
If \(F_{ext}\) = 0 \(\frac{dp}{dt}\) = 0 or P = 0
\(MV_{cm}\) = constant and \(V_{cm}\) = Constant
Collision
It is an isolated event in which two or more colliding bodies exert strong forces on each other for a relatively short time
In all types of collision Total linear momentum is conserved i.e.. initial and final momentum of the system i.e.. Total momentum before collision = Total momentum after collision \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)
If there are absence of any dissipative forces the mechanical system of the system will also remain conserved i.e\(\frac{1}{2}mv_1^2\)+ \(\frac{1}{2}mv_2^2\)= \(\frac{1}{2}mu_1^2\)+ \(\frac{1}{2}mu_2^2\)
Types of collisions
1) Elastic and inelastic collisions
Elastic: Both Kinetic Energy and Momentum are conserved and constant
Inelastic: Only Momentum are conserved and constant and not Kinetic Energy
Perfectly Inelastic Collision: If the approaching particles permanently stick and move together with common velocity
Head on and oblique Collisions
Head on Collisions: The velocity vectors of colliding particles are directed along the line of impact
Oblique Collisions: The velocity vectors of colliding particles are not along the line of impact
Head on Elastic collisions
If two balls of masses m1 and m2 are moving with velocities u1 and u2 in same direction and after collisions the velocities become < v1 and v2then momentum and Kinetic Energy will become
\(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)
\(\frac{1}{2}mv_1^2\)+ \(\frac{1}{2}mv_2^2\)= \(\frac{1}{2}mu_1^2\)+ \(\frac{1}{2}mu_2^2\)
Solving Above two equations we get
v1 = \(\frac{(m_1-m_2)u_1}{m_1 + m_2}\) + \(\frac{2m_2u_2}{ m_1 + m_2}\)
v2 = \(\frac{2m_1u_1}{m_1 + m_2}\) +\(\frac{(m_2-m_1)u_2}{m_1+m_2}\)
Cases
1) If m1 = m2 and subtitling in above equations we get
V1 = U2
V2 = U1
2) If m1 >> m2and subtitling in above equations we get
V1 = 0
V2 = -U2
3) If m1 <<<< m2, u1 = 0 and subtitling in above equations we get
V1 = 2u2
V2 = u2
Head on Inelastic collisions
If two balls of masses m1 and m2 are moving with velocities u1 and u2 in same direction and after collisions the velocities become Same (V)
(m1u1 + m2u2) = (m1 + m2)V
Loss in Kinetic Energy (ΔK)
\(\frac{1}{2}mu_1^2\)+ \(\frac{1}{2}mu_2^2\)= \(\frac{1}{2} (\frac{m_1m_2}{m_1+m_2}) (u_1^2-u_2^2\))
and by solving this we get
ΔK = \(\frac{1}{2}\) \(\frac{m_1m_2}{m_1+m_2}\) (V1 2 – V2 2) and we see that there is loss in KE in inelastic collision
Newton Law of Restitution
When two bodies are in direct impact the speed with which they can separate after impact is usually less than or equal to speed of approach before impact
E = \(\frac{Velocity of separation}{Velocity of approach}\)
E = \(\frac{V_1 – V_2}{U_2 – U_1}\)
V1 = \(\frac{(m_2-em_1)u_2}{(m_1+m_2)}\) +\(\frac{2m_2u_2}{m_1 + m_2}\)
V2 = \(\frac{2m_1u_1}{m_1 + m_2}\) + \(\frac{(m_2-em_1)u_2}{ m_1 + m_2}\)
For Elastic Collision e =1 and for inelastic collision e = 0 and for perfectly inelastic collision e = 0
If M2 >> M1 then after collision velocity of M2 does not change appreciably
If a ball is freely fallen and e is the coefficient of restitution b/w ball and ground then then
Vn = en Vo = en = \(\sqrt{2gh}\)
2)hn = e2n h
3) H = h (\(\frac{1+e^2}{1-e^2}\))
4) T = \(\frac{1+e}{1-e}\) = \(\sqrt\frac{2h}{g}\)
Loss in KE in inelastic collision
\(\frac{1}{2}mu_1^2\)+\(\frac{1}{2}mu_2^2\)=\(\frac{1}{2}\)\(\frac{m_1m_2}{m_1+m_2}\)(V1–V22)(1- e2)
Centre of Mass Law of Conservation of Momentum and Collisions
CategoriesNEET Physics