Motion of Centre of Mass

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JEE Main 2019, 8 April Shift-I

Four particles $A, B, C$ and $D$ with masses $m_{A} = m$ , $m_{B} = 2 m$ , $m_{C} = 3 m$ and $m_{D} = 4 m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles (in $ms^{- 2}$ ) is

\frac{a}{5} (\hat{i} - \hat{j})

(B)

a (\hat{i} + \hat{j})

(C)

zero

(D)

\frac{a}{5} (\hat{i} + \hat{j})

LEVELJEE Main

A body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $\frac{1}{3} M$ and a body $C$ of mass $\frac{2}{3} M$ . The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

(A)

depends on height of breaking

C

(D)

body

B

LEVELJEE Main

Two identical particles move towards each other with velocity $2 v$ and $v$ , respectively. The velocity of centre of mass is

(A)

v

(B)

\frac{v}{3}

(C)

\frac{v}{2}

(D)

zero

JEE Main 2021, 26 Feb Shift-II

LEVELJEE Main

Two masses $A$ and $B$ , each of mass $M$ are fixed together by a massless spring. A force acts on the mass $B$ as shown in figure. If the mass $A$ starts moving away from mass $B$ with acceleration $a$ , then the acceleration of mass $B$ wil be

(A)

\frac{M a - F}{M}

(B)

\frac{M F}{F + M a}

(C)

\frac{F + M a}{M}

(D)

\frac{F - M a}{M}

JEE Advanced 1996

LEVELJEE Main

A small sphere of radius $R$ is held against the inner surface of a larger sphere of radius $6 R$ . The masses of large and small spheres are $4 M$ and $M$ respectively. This arrangement is placed on a horizontal table. There is no friction between any surfaces of contact. The small sphere is now released. Find the coordinates of the centre of the larger sphere when the smaller sphere reaches the other extreme position.

JEE Advanced 2003

LEVELJEE Advanced

Two point masses $m_{1}$ and $m_{2}$ are connected by a spring of natural length $l_{0}$ . The spring is compressed such that the two point masses touch each other and then they are fastened by a string. Then the system is moved with a velocity $v_{0}$ along positive x-axis. When the system reaches the origin the string breaks ( $t = 0$ ). The position of the point mass $m_{1}$ is given by $x_{1} = v_{0} t - A (1 - cos ω t)$ where $A$ and $ω$ are constants. Find the position of the second block as a function of time. Also, find the relation between $A$ and $l_{0}$ .

JEE Advanced (1983)

LEVELJEE Advanced

A block of mass $M$ with a semicircular track of radius $R$ , rests on a horizontal frictionless surface. A uniform cylinder of radius $r$ and mass $m$ is released from rest at the top point $A$ (see fig.). The cylinder slips on the semicircular frictionless track. (a) How far has the block moved when the cylinder reaches the bottom (point $B$ ) of the track ? (b) How fast is the block moving when the cylinder reaches the bottom of the track?

JEE Advanced 1993

LEVELJEE Advanced

A uniform thin rod of mass $M$ and length $L$ is standing vertically along the $Y$ -axis on a smooth horizontal surface, with its lower end at the origin $(0, 0)$ . A slight disturbance at $t = 0$ causes the lower end to slip on the smooth surface along the positive $X$ -axis, and the rod starts falling. (1993) (a) What is the path followed by the centre of mass of the rod during its fall? (b) Find the equation of the trajectory of a point on the rod located at a distance $r$ from the lower end. What is the shape of the path of this point ?

JEE Advanced 2017

LEVELJEE Advanced

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x = 0$ , in a coordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $x$ and the velocity is $v$ . At that instant, which of the following option is/are correct?

* Multiple Correct Options

(A)

The velocity of the point mass

m

v = \frac{2 g R}{1 + \frac{m}{M}}

(B)

The

x

component of displacement of the centre of mass of the block

M

- \frac{m R}{M + m}

(C)

The position of the point mass is

x = - 2 \frac{m R}{M + m}

(D)

The velocity of the block

M

V = - \frac{m}{M} 2 g R

JEE Advanced 1985

LEVELJEE Advanced

A ball of mass $100 g$ is projected vertically upwards from the ground with a velocity of $49 m/s$ . At the same time, another identical ball is dropped from a height of $98 m$ to fall freely along the same path as that followed by the first ball. After some time, the two balls collide and stick together and finally fall to the ground. Find the time of flight of the masses.

JEE Advanced 1982

LEVELJEE Main

Two particles $A$ and $B$ initially at rest, move towards each other by mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2 v$ , the speed of the centre of mass of the system is

(A)

3 v

(B)

v

(C)

1.5 v

Two blocks of masses $10 kg$ and $4 kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of $14 m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is

(A)

30 m/s

(B)

20 m/s

(C)

10 m/s

(D)

5 m/s

JEE Advanced 1997

LEVELJEE Main

An isolated particle of mass $m$ is moving in horizontal plane $(x - y)$ , along the $X$ -axis, at a certain height above the ground. It suddenly explodes into two fragment of masses $m /4$ and $3 m /4$ . An instant later, the smaller fragment is at $y = + 15 cm$ . The larger fragment at this instant is at

(A)

y = - 5 cm

(B)

y = + 20 cm

(C)

y = + 5 cm

(D)

y = - 20 cm

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

A projectile is thrown from a point O on the ground at an angle

4 5^{\circ}

from the vertical and with a speed

52 m/s

. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,

0.5 s

after the splitting. The other part, t seconds after the splitting, falls to the ground at a distance x meters from the point O. The acceleration due to gravity

g = 10 m/s^{2}

Question 1:

The value of t is ___ .

Question 2:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two identical particles move towards each other with velocity 2v and v, respectively. The velocity of centre of mass is

Comprehension Passage

The value of t is ___ .

The value of x is ___ .

Two identical particles move towards each other with velocity $2 v$ and $v$ , respectively. The velocity of centre of mass is