NCERT Class 11 Physics Solutions: Chapter 7 – System of Particles and Rotational Motion Part 1

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Question 7.1:

Give the location of the centre of mass of a

Sphere,

Cylinder,

Ring, and

Cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?

Answer:

Geometric centre: No;

The centre of mass (C.M.) is a point where the mass of a body is supposed to be concentrated. For the given geometric shapes having a uniform mass density, the C.M. lies at their respective geometric centers.

The centre of mass of a body need not necessarily lie within it. For example, the C.M. of bodies such as a ring, a hollow sphere, etc., lies outside the body.

Question 7.2:

In the molecule, the separation between the nuclei of the two atoms is about . Find the approximate location of the of the molecule, given that a chlorine atom is about as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Answer:

The given situation can be shown as:

In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 〖10〗^(-10) m)

HCl Molecule, the Separation between the Nuclei of the Two Atoms.

In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 〖10〗^(-10) m)

Distance between and atoms

Mass of atom

Mass of atom

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

Distance of the centre of mass from the atom

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

Therefore, we can have:

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

Question 7.3:

A child sits stationary at one end of a long trolley moving uniformly with a speed 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 of the (trolley + child) system?

Answer: No change;

Explanation:

The child is running arbitrarily on a trolley moving with velocity . 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.

Question 7.4:

Show that the area of the triangle contained between the vectors and is one half of the magnitude of

Answer:

Consider two vectors and , inclined at an angle , as shown in the following figure:

Consider two vectore and angle shown figure.

Figure Shown Vector and Angle.

Consider two vectore and angle shown figure.

In , we can write the relation:

Question 7.5:

Show that is equal in magnitude to the volume of the parallelepiped formed on the three vectors,

Answer:

A parallelepiped with origin and sides , , and is shown in the following figure:

A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Parallelepiped Formed on the Three Vectors, a,B,C.

A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Volume of the given parallelepiped

Let be a unit vector perpendicular to both and . Hence, and have the same direction.

Volume of the parallelepiped