Write the law of conservation of total linear momentum for the system of particle.
Write the law of conservation of total linear momentum for the system of particle.
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A bomb of mass $10\, kg$ explodes into two pieces of masses $4\, kg$ and $6\, kg$. If kinetic energy of $4\, kg$ piece is $200\, J$. Find out kinetic energy of $6\, kg$ piece
A bomb of mass $10\, kg$ explodes into two pieces of masses $4\, kg$ and $6\, kg$. If kinetic energy of $4\, kg$ piece is $200\, J$. Find out kinetic energy of $6\, kg$ piece
State and establish principle of conservation of mechanical energy.
State and establish principle of conservation of mechanical energy.
A body falls towards earth in air. Will its total mechanical energy be conserved during the fall ? Justify.
A body falls towards earth in air. Will its total mechanical energy be conserved during the fall ? Justify.
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass
$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
A mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision, the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$, respectively. All motion is in one dimension. Then,
A mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision, the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$, respectively. All motion is in one dimension. Then,
- [KVPY 2019]