Centre of Mass

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JEE Main 2019, 8 April Shift-II

A uniform rectangular thin sheet $A B C D$ of mass $M$ has length $a$ and breadth $b$ , as shown in the figure. If the shaded portion $H B GO$ is cut-off, the coordinates of the centre of mass of the remaining portion will be

(\frac{2 a}{3}, \frac{2 b}{3})

(B)

(\frac{5 a}{12}, \frac{5 b}{12})

(C)

(\frac{3 a}{4}, \frac{3 b}{4})

(D)

(\frac{5 a}{3}, \frac{5 b}{3})

LEVELJEE Main

A circular disc of radius $R$ is removed from a bigger circular disc of radius $2 R$ , such that the circumferences of the discs coincide. The centre of mass of the new disc is $α R$ from the centre of the bigger disc. The value of $α$ is

(A)

\frac{1}{3}

(B)

\frac{1}{2}

(C)

\frac{1}{6}

(D)

\frac{1}{4}

JEE Main 2019, 12 Jan Shift-I

LEVELJEE Main

The position vector of the centre of mass $r_{c m}$ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

(A)

r = \frac{13}{8} L \overset{x}{^} + \frac{5}{8} L \overset{y}{^}

(B)

r = \frac{11}{8} L \overset{x}{^} + \frac{3}{8} L \overset{y}{^}

(C)

r = \frac{3}{8} L \overset{x}{^} + \frac{11}{8} L \overset{y}{^}

(D)

r = \frac{5}{8} L \overset{x}{^} + \frac{13}{8} L \overset{y}{^}

JEE Main 2020, 8 Jan Shift-II

LEVELJEE Advanced

As shown in figure, when a spherical cavity (centred at $O$ ) of radius 1 is cut out of a uniform sphere of radius $R$ (centred at $C$ ), the centre of mass of remaining (shaded) part of sphere is at $G$ , i.e. on the surface of the cavity. $R$ can be determined by the equation

(A)

(R^{2} + R + 1) (2 - R) = 1

(B)

(R^{2} + R - 1) (2 - R) = 1

(C)

(R^{2} - R - 1) (2 - R) = 1

(D)

(R^{2} - R + 1) (2 - R) = 1

JEE Main 2021, 22 July Shift-II

LEVELJEE Main

The position of the centre of mass of a uniform semi-circular wire of radius $R$ placed in XY-plane with its centre at the origin and the line joining its ends as X-axis is given by $(0, \frac{x R}{π})$ . Then, the value of $∣ x ∣$ is ......... .

JEE Main 2020

LEVELJEE Main

The centre of mass of a solid hemisphere of radius $8 cm$ is $x cm$ from the centre of the flat surface. Then, value of $x$ is …… .

JEE Main 2020, 7 Jan Shift-I

LEVELJEE Main

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The centre of mass of the system is at a point

(A)

2.0 cm right and 0.9 cm above 1 kg mass

(B)

0.6 cm right and 2.0 cm above 1 kg mass

(C)

1.5 cm right and 1.2 cm above 1 kg mass

(D)

0.9 cm right and 2.0 cm above 1 kg mass

JEE Main 2019, 12 April Shift-II

LEVELJEE Main

Three particles of masses $50 g$ , $100 g$ and $150 g$ are placed at the vertices of an equilateral triangle of side $1 m$ (as shown in the figure). The $(x, y)$ coordinates of the centre of mass will be

(A)

(\frac{3}{4} m, \frac{5}{12} m)

(B)

(\frac{7}{12} m, \frac{3}{8} m)

(C)

(\frac{7}{12} m, \frac{3}{4} m)

(D)

(\frac{3}{8} m, \frac{7}{12} m)

JEE Main 2020, 9 Jan Shift-II

LEVELJEE Main

A rod of length $L$ has non-uniform linear mass density given by $ρ (x) = a + b (\frac{x}{L})^{2}$ , where $a$ and $b$ are constants and $0 \leq x \leq L$ . The value of $x$ for the centre of mass of the rod is at

(A)

\frac{3}{4} (\frac{2 a + b}{3 a + b}) L

(B)

\frac{4}{3} (\frac{a + b}{2 a + 3 b}) L

(C)

\frac{3}{2} (\frac{a + b}{2 a + b}) L

(D)

\frac{3}{2} (\frac{2 a + b}{3 a + b}) L

JEE Main 2020

LEVELJEE Main

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass $4 kg$ . (The coordinates of the same are shown in figure) are

(A)

(1.25 m, 1.50 m)

(B)

(1 m, 1.75 m)

(C)

(0.75 m, 0.75 m)

(D)

(0.75 m, 1.75 m)

JEE Main 2019, 9 Jan Shift-I

LEVELJEE Main

An $L$ -shaped object made of thin rods of uniform mass density is suspended with a string as shown in figure. If $A B = B C$ and the angle is made by $A B$ with downward vertical is $θ$ , then

(A)

tan θ = \frac{2}{3}

(B)

tan θ = \frac{1}{2 3}

(C)

tan θ = \frac{1}{2}

(D)

tan θ = \frac{1}{3}

JEE Main 2015

LEVELJEE Main

Distance of the centre of mass of a solid uniform cone from its vertex is $z_{0}$ . If the radius of its base is $R$ and its height is $h$ , then $z_{0}$ is equal to

(A)

\frac{h ^{2}}{4 R}

(B)

\frac{3 h}{4}

(C)

\frac{5 h}{8}

(D)

\frac{3 h ^{2}}{8 R}

LEVELJEE Advanced

A thin rod of length $L$ is lying along the x-axis with its ends at $x = 0$ and $x = L$ . Its linear density (mass/length) varies with $x$ as $k (\frac{x}{L})^{n}$ , where $n$ can be zero or any positive number. If the position $x_{C M}$ of the centre of mass of the rod is plotted against $n$ , which of the following graphs best approximates the dependence of $x_{C M}$ on $n$ ?

JEE Main 2021, 24 Feb Shift-II

LEVELJEE Main

A circular hole of radius $(\frac{a}{2})$ is cut out of a circular disc of radius $a$ as shown in figure. The centroid of the remaining circular portion with respect to point $O$ will be

(A)

\frac{1}{6} a

(B)

\frac{10}{11} a

(C)

\frac{5}{6} a

(D)

\frac{2}{3} a

LEVELJEE Main

A T shaped object with dimensions shown in the figure, is lying on a smooth floor. A force F is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C.

(A)

\frac{2}{3} l

(B)

\frac{3}{2} l

(C)

\frac{4}{3} l

(D)

l

LEVELJEE Main

Consider a two particle system with particles having masses $m_{1}$ and $m_{2}$ . If the first particle is pushed towards the centre of mass through a distance $d$ , by what distance should the second particle be moved, so as to keep the centre of mass at the same position ?

(A)

\frac{m _{2}}{m _{1}} d

(B)

\frac{m _{1}}{m _{1} + m _{2}} d

(C)

\frac{m _{1}}{m _{2}} d

(D)

d

JEE Main 2020, 02 Sep Shift-II

LEVELJEE Advanced

A square shaped hole of side $l = \frac{a}{2}$ is curved out at a distance $d = \frac{a}{2}$ from the centre $O$ of a uniform circular disc of radius $a$ . If the distance of the centre of mass of the remaining portion from $O$ is $- a / X$ , value of $X$ (to the nearest integer) is ……… .

JEE Advanced (1980)

LEVELJEE Main

A circular plate of uniform thickness has a diameter of $56 cm$ . A circular portion of diameter $42 cm$ is removed from one edge of the plate as shown in figure. Find the position of the centre of mass of the remaining portion.

JEE Advanced 2009

LEVELJEE Main

Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $m$ . The mass of the ink used to draw the outer circle is $6 m$ . The coordinates of the centres of the different parts are : outer circle $(0, 0)$ , left inner circle $(- a, a)$ , right inner circle $(a, a)$ , vertical line $(0, 0)$ and horizontal line $(0, - a)$ . The $y$ - coordinate of the centre of mass of the ink in this drawing is

(A)

\frac{a}{10}

(B)

\frac{a}{8}

(C)

\frac{a}{12}

(D)

\frac{a}{3}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The position vector of the centre of mass rcm​ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centre of mass of a solid hemisphere of radius 8 cm is x cm from the centre of the flat surface. Then, value of x is …… .

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The centre of mass of the system is at a point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in figure) are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Distance of the centre of mass of a solid uniform cone from its vertex is z0​. If the radius of its base is R and its height is h, then z0​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A circular hole of radius (2a​) is cut out of a circular disc of radius a as shown in figure. The centroid of the remaining circular portion with respect to point O will be

A T shaped object with dimensions shown in the figure, is lying on a smooth floor. A force F is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C.

The position vector of the centre of mass $r_{c m}$ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

The centre of mass of a solid hemisphere of radius $8 cm$ is $x cm$ from the centre of the flat surface. Then, value of $x$ is …… .

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass $4 kg$ . (The coordinates of the same are shown in figure) are

Distance of the centre of mass of a solid uniform cone from its vertex is $z_{0}$ . If the radius of its base is $R$ and its height is $h$ , then $z_{0}$ is equal to

A circular hole of radius $(\frac{a}{2})$ is cut out of a circular disc of radius $a$ as shown in figure. The centroid of the remaining circular portion with respect to point $O$ will be