Principles of Mechanics pp 87102  Cite as
System of Particles
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Abstract
In the previous chapters, objects that can be treated as particles were only considered. We have seen that this is possible only if all parts of the object move in exactly the same way An object that does not meet this condition must be treated as a system of particles. Next, we will see that the complex motion of this object or system of particles can be represented by the motion of a point located at the center of mass of the system. The center of mass moves as if all of the mass of the object is concentrated there and as if the net external force acting on the system is applied there (at the center of mass). As well as representing an object by a particle, the concept of the center of mass is used to analyze the motion of many systems such as a system of two colliding blocks (particlelike objects) and the system of two colliding subatomic particles such as the neutron with the nucleus.
6.1 System of Particles
In the previous chapters, objects that can be treated as particles were only considered. We have seen that this is possible only if all parts of the object move in exactly the same way. An object that does not meet this condition must be treated as a system of particles. Next, we will see that the complex motion of this object or system of particles can be represented by the motion of a point located at the center of mass of the system. The center of mass moves as if all of the mass of the object is concentrated there and as if the net external force acting on the system is applied there (at the center of mass). As well as representing an object by a particle, the concept of the center of mass is used to analyze the motion of many systems such as a system of two colliding blocks (particlelike objects) and the system of two colliding subatomic particles such as the neutron with the nucleus.
6.2 Discrete and Continuous System of Particles
6.2.1 Discrete System of Particles
A discrete system of particles is a system in which particles are separated from each other.
6.2.2 Continuous System of Particles
A continuous system of particles is a system where the separation of particles is very small such that it approaches zero. An extended object is a continuous system of particles. Now, consider the skateboarder example mentioned in Sect. 4.3. It has been shown that the system (man\(+\)skateboard) cannot be treated as a particle since different parts of the system move in different ways. By representing the skateboarder as a system of particles its motion can be represented by the motion of its center of mass, hence, the work–energy theorem can be applied to that point. The work done by the force, exerted on the skateboarder by the bar, is not zero because the point of application of that force (which is at the center of mass) has moved.
6.3 The Center of Mass of a System of Particles
6.3.1 Two Particle System
6.3.2 Discrete System of Particles
Example 6.1
Find the center of mass of the system shown in Fig. 6.3 where the three particles have an equal mass of \(m=1\, \mathrm {k}\mathrm {g}.\)
Solution 6.1
Example 6.2
A system of particles consists of three masses \(m_{A}=0.5\) kg, \(m_{B}=2\) kg and \(m_{C}=5\) kg located at \(\mathrm {P}_{\mathrm {A}}(3,1,2)\) , \(\mathrm {P}_{\mathrm {B}}(0,1,2)\) and \(\mathrm {P}_{\mathrm {C}}(1,3,0)\), respectively. Find the position vector of the center of mass of the system.
Solution 6.2
6.3.3 Continuous System of Particles (Extended Object)
6.3.4 Elastic and Rigid Bodies
Example 6.3
A thin rod of length \(L=2\,\mathrm {m}\) has a linear density that increases with x according to the expression \(\lambda (x)=(2x1)\,\mathrm {k}\mathrm {g}/\mathrm {m}\) (see Fig. 6.5). Locate the center of mass of the rod relative to O.
Solution 6.3
Example 6.4
A uniform square sheet is suspended by a uniform rod where they both lie in the same plane as shown in Fig. 6.6. Find the center of mass of the system.
Solution 6.4
Example 6.5
Find the center of mass of the rectangular plate shown in Fig. 6.7. The plate has a uniform surface density \(\sigma .\)
Solution 6.5
 Method 1:$$ x_{cm}=\frac{\int xdm}{M}=\frac{\int x\sigma dA}{\int \sigma dA}=\frac{\int _{y=0}^{b}\int _{x=0}^{a}xdxdy}{\int _{y=0}^{b}\int _{x=0}^{a}dxdy}=\frac{ba^{2}}{2ab}=\frac{a}{2} $$Hence$$ y_{cm}=\frac{\int ydm}{M}=\frac{\int x\sigma dA}{\int \sigma dA}=\frac{\int _{x=0}^{a}\int _{y=0}^{b}ydxdy}{\int _{x=0}^{a}\int _{y=0}^{b}dxdy}=\frac{ab^{2}}{2ab}=\frac{b}{2} $$$$ \mathbf {r}_{cm}=\frac{a}{2}\mathbf {i}+\frac{b}{2}\mathbf {j} $$

Method 2:
Dividing the plate into very thin rods each of mass \(\sigma bdx\) givesSimilarly by dividing the plate into thin horizontal rods each of mass \(\sigma ady\) gives$$ x_{cm}=\frac{\int xdm}{M}=\frac{1}{M}\int x\sigma dA=\frac{1}{M}\bigg (\frac{M}{ab}\bigg )\int _{x=0}^{a}\ xbdx=\frac{1}{a}\bigg [\frac{x^{2}}{2}\bigg ]_{x=0}^{a}=\frac{a}{2} $$and$$ y_{cm}=\frac{\int ydm}{M}=\frac{1}{M}\int y\sigma dA=\frac{1}{M}\bigg (\frac{M}{ab}\bigg )\int _{y=0}^{b}\ aydy=\frac{1}{b}\bigg [\frac{y^{2}}{2}\bigg ]_{y=0}^{b}=\frac{b}{2} $$$$ \mathbf {r}_{cm}=\frac{a}{2}\mathbf {i}+\frac{b}{2}\mathbf {j} $$
Example 6.6
An object of uniform surface density \(\sigma \) and mass M has the shape shown in Fig. 6.8 (half of an ellipse). Find the center of mass of the object.
Solution 6.6
Example 6.7
Determine the center of mass of the cylindrical shell shown in Fig. 6.9. The shell has a uniform surface density \(\sigma .\)
Solution 6.7
Example 6.8
A boy standing on a smooth ice surface wants to fetch a container that is at a distance of 10 \(\mathrm {m}\) away from him. To do that, he throws a rope around the container and start to pull. Because the surface is smooth, both the boy and the container will move until they meet. If the masses of the boy and of the container are 40 kg and 70 kg respectively, how far will the container move when the boy has moved a distance of 2 \(\mathrm {m}\)?
Solution 6.8
Example 6.9
A boy is standing at the rear of a boat as shown in Fig. 6.11. The masses of the boy and of the boat are 45 kg and 80 kg respectively Find the distance that the boat would move relative to the origin if the boy moves a distance of lm from the rear of the boat (the length of the boat is \(5\,\mathrm {m}\)).
Solution 6.9
6.3.5 Velocity of the Center of Mass
6.3.6 Momentum of a System of Particles
Example 6.10
Two particles of masses \(m_{1}=1\) kg and \(m_{2}=2\) kg have position vectors given by \(\mathbf {r}_{1}=(2t\mathbf {i}4\mathbf {j})\,\mathrm {m}\) and \(\mathbf {r}_{2}=(5t\mathbf {i}2t\mathbf {j})\,\mathrm {m}\) respectively where t is time. Determine the velocity and linear momentum of the center of mass of the two particle system at any time and at \(t=1\,\mathrm {s}.\)
Solution 6.10
6.3.7 Motion of a System of Particles
6.3.8 Conservation of Momentum
6.3.9 Angular Momentum of a System of Particles
6.3.10 The Total Torque on a System
6.3.11 The Angular Momentum and the Total External Torque
6.3.12 Conservation of Angular Momentum
6.3.13 Kinetic Energy of a System of Particles
6.3.14 Work
6.3.15 Work–Energy Theorem
6.3.16 Potential Energy and Conservation of Energy of a System of Particles
6.3.17 Impulse
6.4 Motion Relative to the Center of Mass
6.4.1 The Total Linear Momentum of a System of Particles Relative to the Center of Mass
6.4.2 The Total Angular Momentum About the Center of Mass
6.4.3 The Total Kinetic Energy of a System of Particles About the Center of Mass
6.4.4 Total Torque on a System of Particles About the Center of Mass of the System
Example 6.11
Two particles of masses \(m_{1}=1\) kg and \(m_{2}=2\) kg are moving in the xy plane. Their position vectors relative to the origin are \(\mathbf {r}_{1}=(t^{2}\mathbf {i}2t\mathbf {j})\,\mathrm {m}\) and \(\mathbf {r}_{2}=(3t\mathbf {i}+\mathbf {j})\,\mathrm {m}\) where t is time. Find: (a) the total angular momentum of the system; the total external torque acting on the system; and the total kinetic energy of the system all relative to the origin at any time; (b) repeat (a) relative to the center of mass.
Solution 6.11
\(\displaystyle \varvec{\tau '}=\frac{d\mathbf {L}'}{dt}=\bigg (\bigg (\frac{16}{3}t+\frac{4}{3}\bigg )\mathbf {k}\bigg )\,\mathrm {N.m}\)
Example 6.12
Two particles of equal mass m are rotating about their center of mass with a constant speed v as in Fig. 6.13. If they are separated by a distance 2d, find the total angular momentum of the system.
Solution 6.12
6.4.5 Collisions and the Center of Mass Frame of Reference
Example 6.13
A rocket is projected vertically upward and explodes into three fragments of equal mass when it reaches the top of its flight at an altitude of 40 \(\mathrm {m}\) (see Fig. 6.16). If the two fragments land to the ground after 3 \(\mathrm {s}\) from the explosion, find the time it takes the third fragment to hit the ground.
Solution 6.13
\(29.4t^{2}+160t+63.6=0\)
Thus, \(t=2.3\,\mathrm {s}.\)
Example 6.14
Find the center of mass of the Earth–Moon System and describe its motion around the sun.
Solution 6.14
Example 6.15
Describe the motion of a rocket in space using the law of conservation of momentum.
Solution 6.15
 1.
Find the coordinate of the center of mass of the system shown in Fig. 6.20.
 2.
Find the center of mass of a uniform plate bounded by \(y=0.24x^{2}+6\) and the \(\mathrm {x}\)axis from \(x=5\) to \(x=5\,\mathrm {m}.\)
 3.
Find the center of mass of the homogeneous sheet shown in Fig. 6.21.
 4.
Find the center of mass of the homogeneous sheet shown in Fig. 6.22.
 5.
Find the center of mass of a uniform solid circular cone of radius a and height h.
 6.
Find the center of mass of a uniform solid hemisphere of radius R.
 7.
Two masses initially at rest are located at the points shown in Fig. 6.23. If external forces act on the particles as in Fig. 6.23, find the acceleration of the center of mass.
 8.
A projectile of mass 15 kg is fired from the ground with an initial velocity of 12 \(\mathrm {m}/\mathrm {s}\) at an angle of \(45^{\mathrm {o}}\) to the horizontal. 1 second later, the projectile explodes into two fragments A and B. If immediately after explosion, fragment A has a mass of 5 kg and a speed of 5 \(\mathrm {m}/\mathrm {s}\) at an angle of \(30^{\mathrm {o}}\) to the horizontal, find the velocity of fragment B (assuming air resistance is neglected).
 9.
Two boys of masses 45 and 40 kg are standing on a boat of mass 150 kg and length 5 \(\mathrm {m}\) as in Fig. 6.24. The boat is initially lm from the pier. Assuming that there is no friction between the boat and the water, find the distance moved by the boat when the two meet at the middle of the boat.
 10.
Two particles of masses \(m_{1}=3\) kg and \(m_{2}=5\) kg are moving relative to the lab frame with velocities of 10 \(\mathrm {m}/\mathrm {s}\) along the \(\mathrm {y}\)axis and 15 \(\mathrm {m}/\mathrm {s}\) at an angle of \(30^{\mathrm {o}}\) to the \(\mathrm {x}\)axis. Find (a) the velocity of their center of mass (b) the momentum of each particle in the center of mass frame (c) the total kinetic energy of the particles relative to the lab frame and relative to the center of mass frame.
 11.
Two particles of masses \(m_{1}=1\) kg and \(m_{2}=2\) kg are moving relative to the lab frame with velocities of \(\mathbf {v}_{1}=2\mathbf {i}3\mathbf {j}+\mathbf {k}\) and \(\mathbf {v}_{2}=7\mathbf {i}+\mathbf {j}2\mathbf {k}\). If at a certain instant they are located at \((1,1,2)\) and (3, 0, 1) , find the angular momentum of the system relative to the origin and relative to the center of mass.
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