Four particles of masses m_1 = 2m,m_2 = 4m,m_ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the side length of the square be $2a$. Place the centre of the square at the origin $(0,0)$.The coordinates of the corners are: $m_1$ at $(-a,

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) Let the side length of the square be $2a$. Place the centre of the square at the origin $(0,0)$.The coordinates of the corners are: $m_1$ at $(-a, -a)$,$m_2$ at $(a, -a)$,$m_3$ at $(a, a)$,and $m_4$ at $(-a, a)$.The $x$-coordinate of the centre of mass is given by:$X_{cm} = \frac{m_1(-a) + m_2(a) + m_3(a) + m_4(-a)}{m_1 + m_2 + m_3 + m_4} = 0$Substituting the given masses:$2m(-a) + 4m(a) + m(a) + m_4(-a) = 0$$-2ma + 4ma + ma - m_4a = 0$$3ma - m_4a = 0 \implies m_4 = 3m$The $y$-coordinate of the centre of mass is given by:$Y_{cm} = \frac{m_1(-a) + m_2(-a) + m_3(a) + m_4(a)}{m_1 + m_2 + m_3 + m_4} = 0$Substituting the given masses:$2m(-a) + 4m(-a) + m(a) + m_4(a) = 0$$-2ma - 4ma + ma + m_4a = 0$$-5ma + m_4a = 0 \implies m_4 = 5m$Since the required values for $m_4$ to satisfy both $X_{cm}=0$ and $Y_{cm}=0$ are different ($3m$ and $5m$),it is impossible for the centre of mass to be at the centre of the square for any value of $m_4$ given these specific masses $m_1, m_2, m_3$.

Four particles of masses $m_1 = 2m$,$m_2 = 4m$,$m_3 = m$,and $m_4$ are placed at the four corners of a square. What should be the value of $m_4$ so that the centre of mass of all the four particles is exactly at the centre of the square?

Similar Questions

Particles of masses $m, 2m, 3m, \ldots, nm$ grams are placed on the same line at distances $l, 2l, 3l, \ldots, nl$ cm from a fixed point. The distance of the centre of mass of the particles from the fixed point in centimetres is:

Three rods of the same mass are placed as shown in the figure. The coordinates of the centre of mass of the system are:

$A$ cricket bat is cut at the location of its centre of mass as shown in the figure. Then:

In an $HCl$ molecule,the distance between the nuclei of the two atoms is $1.27 \ \mathring A$. The $Cl$ atom is approximately $35.5$ times heavier than the $H$ atom. The center of mass of this molecule will be at a distance of approximately ....... $\mathring A$ from the center of the $H$ atom.

Write the meaning of homogeneous bodies.

Vedclass Test Series

Exam Paper Generator

Online Exam Module