Answer

Verified

345.3k+ views

**Hint:**The term "density" refers to the amount of mass per unit of volume. An object's average density is proportional to its total mass divided by its total volume. An object composed of a more dense material (such as iron) would have less volume than one made of a less dense substance of comparable mass (such as water).

**Complete step by step answer:**

The centre of mass of a mass distribution in space (also known as the equilibrium point) is the special point in physics where the weighted relative location of the distributed mass amounts to zero. A force may be applied to this point to induce a linear acceleration without causing an angular acceleration.

We know that

$\text{Mass} = \dfrac{\text{Density}}{\text{Volume}}$

If the density of cone is$\rho $, then mass of cone = $ = {\text{ density }} \times {\text{ volume }} = \dfrac{{\rho 1}}{{3\pi {{(2{\text{R}})}^2}}}$

i.e, ${m_1} = \dfrac{{16}}{{3\pi {{({\text{R}})}^3}\rho }}$

Hence, This mass is concentrated at a height from

${\text{O}} = \dfrac{h}{4} = \dfrac{{4R}}{h}$

From the above calculations

$\text{Mass of sphere} = {m_2} \\

\Rightarrow \text{Mass of sphere} = {\text{ density }} \times {\text{ volume }} \\

\Rightarrow \text{Mass of sphere} = \dfrac{{12\rho 4}}{{3\pi {{({\text{R}})}^3}}} \\

\Rightarrow \text{Mass of sphere} = 16\pi {{\text{R}}^3}\rho $

This mass is actually concentrated at ${O_2}$, i.e. at a distance of \[4R + R = 5R\] from O.Hence centre of mass of whole system will be

${{\text{Y}}_{{\text{cm}}}} = \dfrac{{{{\text{m}}_1}{{\text{y}}_1} + {{\text{m}}_2}{{\text{y}}_2}}}{{\;{{\text{m}}_1} + {{\text{m}}_2}}} \\$

Here ${y_1} = R{\text{ and }}{y_2} = 5R$

${Y_{cm}} = \dfrac{{mR + 3m(5R)}}{{m + 3m}} \\$

$\Rightarrow {Y_{cm}} = \dfrac{{16{\text{mR}}}}{{4\;{\text{m}}}} \\ $

$\therefore {Y_{cm}} = 4R$

**Hence option C is correct.**

**Note:**When formulated with respect to the centre of mass, physics calculations are sometimes simplified. It is a fictional point at which an object's entire mass can be believed to be condensed in order to visualise its motion. To put it another way, the centre of mass is the particle counterpart to a given object when Newton's laws of motion are applied.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE