A system of particles in motion has mass center G as shown in the figure. The particle i has mass m_{i} and its position with respect to a fixed point O is given by the position vector r_{i}. The position of the particle with respect to G is given by the vector ρ_{i}. The time rate of change of the angular momentum of the system of particles about G is

(The quantity \({\ddot \rho _i}\) indicates the second derivative of ρ_{i} with respect to time and likewise for r_{i})

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PY 10: GATE ME 2016 Official Paper: Shift 2

Option 2 : \(\sum_i \rho_i \times m_i \ddot{r}_i\)

CT 1: Ratio and Proportion

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30 Mins

**Angular momentum is the moment of linear momentum about given point. The moment of linear momentum is obtained by product of linear momentum & distance perpendicular from the given point. The time rate of change of angular momentum of a system gives toque.**

\(T = \frac{{d\vec L}}{{dt}}\)

\({T _i} = {\vec r_{perpendicular}} \times \vec F\)

\({r_{perpendicular}} = \rho_i\)

\(\vec{F}=m_i a_i = m_i \ddot{r}_i\)

\(\therefore \ \tau_i = \rho_i \times m_i \ddot{r}_i\)

For complete rigid body

\(\sum \tau_i = \sum_i \rho_i \times m_i \ddot{r}_i\)