Conservation of Linear Momentum

Conservation of Linear Momentum

For two particles

Consider a two body system of masses \(m_{1}\) and \(m_{2}\) moving along with velocities \(u_{1}\) and \(u_{2}\) respectively. Suppose, the bodies collide for a very small interval \(\Delta t\) and exert impulsive forces on each other. The force exerted by\(m_{1}\) on\(m_{2}\) is,\begin{align*} F_{12}=m_{2}\frac{v_{2}-u_{2}}{\Delta t}\end{align*} Similarly, force exerted by \(m_{2}\) on \(m_{1}\) is,\begin{align*} F_{21}=m_{1}\frac{v_{1}-u_{1}}{\Delta t}\end{align*} These forces are action and reaction pairs and are thus equal an opposite in direction. ie. \(F_{12}=-F_{21}\). So,\begin{align*} m_{2}\frac{v_{2}-u_{2}}{\Delta t}=-m_{1}\frac{v_{1}-u_{1}}{\Delta t}\end{align*} Hence, total initial momentum is equal to the total final momentum. Thus, linear momentum of a system remains conserved but only in the absence of external forces. For explosive systems, since,\begin{align*} m_{1}u_{1}+m_{2}u_{2}=0\end{align*}\begin{align*} or,m_{1}v_{1}+m_{2}v_{2}=0\end{align*}\begin{align*} or,v_{1}=(-\frac{m_{2}}{m_{1}})v_{2}\end{align*} where, negative sign inidcates opposite direction of velocity.

For a system of particles

Consider an isolated system of n number of particles having masses \(m_{1}, m_{2}, ...m_{n}\) moving with velocities \(v_{1}, v_{2}, ...v_{n}\) respectively. The total momentum of the system is,\begin{align*} p=p_{1}+p_{2}+...p_{n}\end{align*}\begin{align*} or, p=m_{1}v_{1}+m_{2}v_{2}+...m_{n}v_{n}\end{align*} Differentiating with respect to time,\begin{align*} \frac{\mathrm{dP} }{\mathrm{d} t}=m_{1}\frac{\mathrm{dv_{1}} }{\mathrm{d} t}+m_{2}\frac{\mathrm{dv_{2}} }{\mathrm{d} t}+...m_{n}\frac{\mathrm{dv_{n}} }{\mathrm{d} t}\end{align*} From Newton's second law of motion, the force acting on the system is,\begin{align*} \frac{\mathrm{dP} }{\mathrm{d} t}=F_{1}+F_{2}+...F_{n}\end{align*} A system is said to be isolated only when net force acting on the system is zero ie. \(F_{e}=F_{1}+F_{2}+...F_{n}=0\) So, \begin{align*} \frac{\mathrm{dP} }{\mathrm{d} t}=0 \end{align*} \begin{align*} or, P=constant \end{align*}\begin{align*} or, P_{i}=P_{f} \end{align*} Hence, linear momentum remains conserved for an isoated system of particles.

Centre of mass and centre of gravity

Centre of mass is a point at which total mass of the system is supposed to be concentrated. If we apply a force at the centre of mass of a body, it suffers translational motion only. If \(g\) can't be assumed constant over the whole of the body(perhaps because the body is very tall), a centre of mass and centre gravity do not coincide but they do coincide in the uniform gravitational field. The CM is the point inside the distribution of masses in space that has the property that the weighted position vectors relative to this point sum to zero. In case of a system of n particles with masses \(m_{1}, m_{2}, ...m_{n}\), the co-ordinate of the position of the centre of mass satisfy the condition,\begin{align*} \sum_{1}^{n}m_{i}(r_{i}-R)=0\end{align*} where, R is the position of CM of the system.\begin{align*} R\sum_{1}^{n}m_{i}=\sum_{1}^{n}m_{i}r_{i}\end{align*}\begin{align*} or, MR=\sum_{1}^{n}m_{i}r_{i}\end{align*} where, \(M=\sum_{1}^{n}m_{i}\) is the total mass of the system.

Motion of Centre of Mass

Velocity of centre of mass in lab frame is,\begin{align*} v=\frac{\sum_{1}^{n}n_{k}v_{k}}{M}\end{align*}\begin{align*} or, Mv=\sum_{1}^{k}m_{k}v_{k}\end{align*} ie. total momentum of the system is same as the momentum of the centre of mass in lab frame of reference. So, in order to study the motion of a system, the study of the centre of mass is sufficient. Differentiating above equation with respect to k,\begin{align*} M\frac{\mathrm{dv} }{\mathrm{d} t}=\sum_{1}^{k}m_{k}\frac{\mathrm{dv_{k}} }{\mathrm{d} t} \end{align*}\begin{align*} or, Ma=\sum_{1}^{k}m_{k}a_{k}\end{align*}\begin{align*} Ma=F_{k} \end{align*} ie. effect of resultant force on the system is the same as the effect on the CM ie. \(F=Ma=\sum_{1}^{k}F_{k}\)

Frame of References

A system of a co-ordinate axis which is used to define the position of a particle or an event in two or three-dimensional space is known as the frame of reference should be rigid ie. the co-ordinate axis should not change due to the application of external forces. The simplest frame of reference is the Cartesian system of co-ordinates. A frame of reference with an additional co-ordinate time is known as the sapce-time frame of reference. Inertial frame of reference obey Newton's laws and it is non- accelerating frame of reference. An example of it is the CM frame. The CM frame is also known as zero momentum frame of reference.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.