System of Particles and Rotational Motion

Centre of Mass and Its Motion


  • Centre of mass of a body or system of particles is the point at which the whole mass of the body (or) system supposed to be concentrated.
  • The motion of centre of mass represents the motion of the whole body
  • There may or may not be mass at centre of mass
  • Centre of mass is an imaginary point
  • The moments about the centre of mass is same m1x1 = m2x2
  • The centre of mass of heavier and lighter mass system the centre of mass lies nearer to the heavier mass.
  • \tt X_{cm}=\frac{m_{1}x_{1}+m_{2}x_{2}+...+\ m_{n}x_{n}}{m_{1}+m_{2}+...+\ m_{n}}
  • \tt Y_{cm}=\frac{m_{1}y_{1}+m_{2}y_{2}+...+\ m_{n}y_{n}}{m_{1}+m_{2}+...+\ m_{n}}
  • \tt Z_{cm}=\frac{m_{1}z_{1}+m_{2}z_{2}+...+\ m_{n}z_{n}}{m_{1}+m_{2}+...+\ m_{n}}
  • \tt r_{cm}=\frac{m_{1}r_{1}+m_{2}r_{2}+...+\ m_{n}r_{n}}{m_{1}+m_{2}+...+\ m_{n}}
  • Relative to centre of mass \tt m_{1}\overline{r_{1}}+m_{2}\overline{r_{2}}+...+m_{n}\overline{r_{n}}=0
  • For Continuous mass distribution \tt X_{cm}=\frac{\int x\ dm }{\int dm}
  • For Continuous mass distribution \tt Y_{cm}=\frac{\int y\ dm }{\int dm}
  • For Continuous mass distribution \tt Z_{cm}=\frac{\int z\ dm }{\int dm}
  • For Uniform shaped and regular bodies centre of mass lies at their Geometric centre.
  • For Non uniform and irregular shaped bodies centre of mass lies outside the body.
  • Two circular dices of radii “r1 and r2” of same material are kept in contact \tt X_{cm}=\frac{r_2^2\left(r_{1}+r_{2}\right)}{r_1^2+r_2^2}
  • Two spheres of radii ‘r1’ and ‘r2’ of same material are kept in contact \tt X_{cm}=\frac{r_3^3\left(r_{1}+r_{2}\right) }{r_1^3+r_2^3}
  • Two rods of length L1 and L2 joined as T shape. \tt X_{cm}=\frac{L_2^2}{2\left(L_{1}+L_{2}\right)}
  • Two cylindrical rods kept in contact along their length \tt X_{cm} = \frac{r_2^2\ L_{2}}{\left(r_2^2\ L_{1}+r_2^2\ L_{2}\right)} \left[\frac{L_{1}+L_{2}}{2}\right]
  • If Δx1, Δx2 are shift in positions of 1st and 2nd particles then \tt \Delta X_{cm}=\frac{\pm\ m_{1}\Delta\ x_{1}\pm\ m_{2}\Delta\ x_{2}}{m_{1}+m_{2}} + when shifted along +ve x direction − when shifted along –ve x direction
  • Similarly \tt \Delta Y_{cm}=\frac{\pm\ m_{1}\Delta\ y_{1}\pm\ m_{2}\Delta\ y_{2}}{m_{1}+m_{2}}
  • Velocity of centre of mass \tt V_{cm}=\frac{m_{1}\nu_{1}+\ m_{2}\nu_{2}+...+\ m_{n}\nu_{n}}{m_{1}+m_{2}+...+\ m_{n}}
  • M Vcm = P1 + P2 + ---- + Pn {P1 = Momentum}
  • Two particles moving perpendicular \tt V_{cm}=\sqrt{\frac{m_1^2 v_1^2+m_2^2 v_2^2}{\left(m_{1}+m_{2}\right)^{2}}}
  • Acceleration of centre of mass = \tt a_{cm}=\frac{m_{1}a_{1}+\ m_{2}a_{2}\ +...+\ m_{n}a_{n}}{m_{1}+m_{2}+...+m_{n}}
  • \tt M\ \overline{acm}=F\ external
  • Centre of mass can be accelerated only by a net external force.
  • Internal forces cannot accelerate centre of mass
  • In absence of external forces velocity of cm = O
  • Shift in centre of mass when a small portion of mass is removed \tt x\ shift=\frac{m_{removed}}{m_{total}-m_{removed}}
  • When a circular disc or radius ‘r’ is removed from a circular disc or radius ‘R’. \tt Shift=\frac{r^{2}d}{R^{2}-r^{2}}
  • If a circular portion of diameter “R” is removed from a circular disc of radius R. X shift = \tt \frac{R}{6}
  • If centre of removed plate and original plate coincide then shift is zero.
  • When a sphere of radius ‘r is removed from a sphere of radius R, \tt Shift=\frac{r^{3}d}{R^{3}-r^{3}}
  • When a triangular portion is removed from a square plate of side ‘a’. \tt X_{shift}=\frac{a}{9}

  • For a rectangular plate \tt X_{shift}=\frac{a}{9}

  • When & shell in flight Explodes acm = g downwards.
  • The centre of mass of all fragments will continue to move along the same tragatery as fragments in space.
  • A man of mass ‘m’ walks a distance ‘L’ on the boat & boat is displaced in opposite direction \tt X=\frac{mL}{M+m}.
  • Distance walked by the man relative to shore (or) water is (L – X).
  • When two masses more under mutual force of attraction towards each other \tt \overline{V}_{cm}=0\ and\ \overline{a}_{cm}=0
  • \tt m_{1}\overline{v}_{1}=-m_{2}\overline{v}_{2}   ( F Ext=0)
    \tt m_{1}\overline{a}_{1}=-m_{2}\overline{a}_{2}   ( F Ext=0)
  • Papper’s 1st Law volume V = 2π Xc × S ( S = Surface area)
  • Papper’s 2nd Law Area S = 2π Xc × Πr
  • Centre of mass of uniform rod = L/2.
  • Centre of mass of semi circular ring = 2R/ π.
  • Centre of mass of a semi circular disc of radius R from centre is 4R/3π.
  • Centre of mass of a quadrant of a circle of radius R from centre is \tt 4R/\pi\sqrt{2}
  • Centre of mass of a solid semi sphere of radium ‘R’ from centre is 3R/8.
  • Centre of mass of a hollow semi sphere of radium R is \tt \frac{R}{2}.
  • Centre of mass of a solid cone is h/4 from base.
  • Centre of mass of a hollow cone is h/3 from base.
  • Centre of mass of a solid sphere is R from base.

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1. The position of centre of mass from m_{1} = \frac{m_{2}d}{m_{1} + m_{2}}

2. Position of centre of mass from m_{2} = \frac{m_{1}d}{m_{1} + m_{2}}

3. If position vectors of particles of masses m1 and m2 are r1 and r2 respectively, then \tt r_{CM} = \frac{m_{1}r_{1} + m_{2}r_{2}}{m_{1} + m_{2}}

4. If in a two particle system, particles of masses m1 and m2 are moving with velocities v1 and v2 respectively, then velocity of the centre of mass \tt v_{CM} = \frac{m_{1}v_{1} + m_{2}v_{2}}{m_{1} + m_{2}}

5. If accelerations of the particles are a1and a2 respectively, then acceleration of the centre of mass \tt a_{CM} = \frac{m_{1}a_{1} + m_{2}a_{2}}{m_{1} + m_{2}}

6. Centre of mass of a uniform semicircular ring of radius R lies at a distance of \tt h = \frac{2R}{\pi} from its centre; on the axis of symmetry as shown in the figure.

7. Centre of mass of a uniform semicircular disc of radius R lies at a distance of \tt h = \frac{4R}{3\pi} from the centre on the axis of symmetry as shown in figure.

8. Centre of mass of a hemispherical shell of radius R lies at a distance of \tt h = \frac{R}{2} from its centre on the axis of symmetry as shown in figure.

9. Centre of mass of a solid hemisphere of radius R lies at a distance of \tt h = \frac{3R}{8} from its centre on the axis of symmetry as shown in the figure.

10. Velocity of centre of mass \tt \vec{v}_{CM} = \frac{m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} + .... + m_{N}\vec{v}_{N}}{m_{1} + m_{2} + .... + m_{N}} = \frac{\sum_{i = 1}^{N}m_{i}\vec{v}_{i}}{\sum_{i = 1}^{N}}m_{i} = \frac{\sum_{i = 1}^{N}m_{i}\vec{i}}{M}

11. Acceleration of centre of mass \tt \vec{a}_{CM} = \frac{m_{1}\vec{a}_{1} + m_{2}\vec{a}_{2} + .... + m_{N}\vec{a}_{N}}{m_{1} + m_{2} + .... + m_{N}} = \frac{\sum_{i = 1}^{N}m_{i}\vec{a}_{i}}{\sum_{i = 1}^{N}m_{i}} = \frac{\sum_{i = 1}^{N}m_{i}\vec{a}_{i}}{M}