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    System of Particles and Rotational Motion
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    Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c. 
    A parallelepiped with origin O and sides a, b, and c is shown in the following figure. 


                    

     Hence proved. 

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    Answer
    In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10–10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus. 

       
                         

    Distance between H and Cl atoms = 1.27 Å 

    Mass of H atom = m 

    Mass of Cl atom = 35.5m 

    Let the centre of mass of the system lie at a distance x from the Cl atom. 

    Distance of the centre of mass from the H atom = (1.27 – x)

    Let us assume that the centre of mass of the given molecule lies at the origin.

    Therefore, we can have

     
               m(1.27 – x) + 35.5mx =  0 

                                  1.27 - x = -35.5x 

    ∴                                       x =

                                               =  -0.037 Å 

    Here, the negative sign indicates that the centre of mass lies at the left of the molecule.

    Hence, the centre of mass of the HCl molecule lies 0.037Å from the Cl atom. 
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    Answer
    A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?
     
    The child is running arbitrarily on a trolley moving with velocity v. However, the running of the child will produce no effect on the velocity of the centre of mass of the trolley.


    This is because the force due to the boy’s motion is purely internal. Internal forces produce no effect on the motion of the bodies on which they act. Since no external force is involved in the boy–trolley system, the boy’s motion will produce no change in the velocity of the centre of mass of the trolley.
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    Answer
    Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.
    Consider two vectors OK = vector |a| and OM = vector |b|, inclined at an angle θ, as shown in the following figure.
                             

     


    Area of the triangle contained between the vectors a and b is one-half of the magnitude of a x b. 

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    Answer
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    Give the location of the centre of mass of a
    (i) sphere,
    (ii) cylinder,
    (iii) ring, and
    (iv) cube, each of uniform mass density.
    Does the centre of mass of a body necessarily lie inside the body?
     

    In all the four cases, as the mass density is uniform, the centre of mass is located at their respective geometrical centres.

    No, it is not necessary that the centre of mass of a body should lie on the body.

    For example, in the case of a circular ring, centre of mass is at the centre of the ring, where there is no mass. 
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