Question 47: Centre of Mass and Rotational Motion

Question

Given below are two statements :
Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.
Statement II : The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference.
In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

For a mechanical system consisting of $n$ discrete particles, the total kinetic energy ( $K_{	ext {total }}$ ) of the system is simply the algebraic sum of the individual kinetic energies of all the constituent particles.
If particle $i$ has mass $m_i$ and velocity $v_i$ relative to a given frame of reference, the total kinetic energy is:
$$ \mathrm{K}_{	ext {total }}=\sum \frac{1}{2} \mathrm{~m}_{\mathrm{i}} \mathrm{v}_{\mathrm{i}}^2 $$
Therefore, Statement I is true.
The total kinetic energy of a system of particles is equal to the sum of translational kinetic energy of the centre of mass and internal kinetic energy.
Translational kinetic energy of the centre of mass is the kinetic energy the system would have if all its mass ( $\mathrm{M}_{	ext {total }}$ ) were concentrated at the center of mass and moved with the velocity of the center of mass ( $\mathrm{V}_{\mathrm{cm}}$ ) relative to the origin. $\left(\mathrm{K}_{\mathrm{cm}}=\frac{1}{2} \mathrm{M}_{	ext {total }} \mathrm{V}_{\mathrm{cm}}^2ight)$
Internal kinetic energy is the sum of the kinetic energies of all individual particles measured with respect to the centre of mass frame
$$ \mathrm{K}_{	ext {internal }}=\sum \frac{1}{2} \mathrm{~m}_{\mathrm{i}} \mathrm{v}_{\mathrm{iC}}^2 $$
So, the total kinetic energy is
$$ \mathrm{K}_{	ext {total }}=\frac{1}{2} \mathrm{M}_{	ext {total }} \mathrm{V}_{\mathrm{cm}}^2+\sum_\limits{\mathrm{i}=1} \frac{1}{2} \mathrm{~m}_{\mathrm{i}} \mathrm{v}_{\mathrm{iC}}^2 $$
Therefore, Statement II is also true. Hence, the correct option is (D).

Answer

Both Statement I and Statement II are false

Answer

Statement I is false but Statement II is true

Answer

Statement I is true but Statement II is false

Answer

Both Statement I and Statement II are true

Centre of Mass and Rotational Motion

Choose the correct answer: