Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar

Chapter 6

Center of Mass and Collision - all with Video Answers

Educators

Section 1

Section A

Problem 1

The position vectors of three particles of mass $m_{1}=1 \mathrm{~kg}, m_{2}=2 \mathrm{~kg}$ and $m_{3}=3 \mathrm{~kg}$ are $\vec{r}_{1}-(\hat{i}|4 \hat{j}| \hat{k}) m, \vec{r}_{2}-(\hat{i}|\hat{j}| \hat{k}) m$ and $\vec{r}_{3}-(2 \hat{i} \quad \hat{j} \quad 2 \hat{k})$ respectively. Find the position vec-
tor of their centre of mass.

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Problem 2

Locate the centre of mass of a uniform straight rod of mass $m$ and length $L$.

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Problem 3

Locate the c.m. of a straight rod of length $L$ having linear mass density $\lambda=A s(A$ is a positive constant and $s$ is the distance from left end)

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Problem 4

$\Lambda$ thin rod of length $L$ is lying along the $x$ -axis with its end at $x=0$ and $x=L$. Its lincar mass
densily varics with $x$ as $K\left(\frac{x}{L}\right)^{n} ;$ wherc, $n$ can be zero or any positive number.

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Problem 5

Locate the c.m, of a uniform circular arc shaped rod radius $R$ and it substends an angle $\theta_{\mathrm{a}}$ at its centre.

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Problem 6

Locate the c.m. of a sector of uniform circular disc of radius $R$ and of mass $m$ and il substends an angle $\theta_{0}$ at its cenire.

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Problem 7

Locate the c.m. of a uniform semi circular disc of mass $m$ and radius $R$.

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Problem 8

Locate the c.m. of a uniform circular disc of mass $m$ and radius $R$.

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Problem 9

Locate the c.m. of a circular disc of radius $R$ and surface mass density $\sigma=(A r)$ where, $A$ is positive constant and $r$ is the distance from the centre.

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Problem 10

Locate the c.m. of a uniform rectangular lamina or rectangular disc of mass $m$ and area $(l \times b)$.

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Problem 11

Locate the c.m. of a rectangular lamina or rectangular disc having surface mass density $\sigma-\sigma_{o}\left(\frac{x y}{a b}\right)$ where the area of the plate is $(a \times b)$ as in figure.

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Problem 12

Locate the $\mathrm{c}, \mathrm{m}$, of a uniform hollow cylinder of base radius $R$, height $H$ and mass $m$.

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Problem 13

Locate the c.m. of a uniform solid cylinder of base radius $R$, height $H$ and mass $m$.

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Problem 14

Locale the c.m. of a uniform solid cone of base radius $R$, height $H$ and mass $m$.

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Problem 15

Locate the $\mathrm{c}, \mathrm{m}$, of a uniform solid sphere of radius $R$ and mass $m$.

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Problem 16

Locate the c.m, of a uniform hollow sphere of radius $R$ and mass $m$.

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Problem 17

Locate the c.m. of a uniform cuboid of mass $m$ and having dimensions $(l \times b \times h)$.

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Problem 18

From a uniform disc of radius $R$, a circular hole of radius $R / 2$ is cut. The centre of the hole is at $R / 2$ from the centre of the original disc. Locate the centre of mass of the resulting flat body.

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Problem 19

In the figure a uniform disc of radius $R$, from which a hole of radius $R / 2$ has been cut out from left of the centre and is placed on right of the centre of disc. Find the $C . M$. of the resulting disc.

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Problem 20

Find the $c . m$, of a uniform $L$ -shapod lamina (a thin Ilat platc) with dimcnsions as shown in figurc. The mass of lamina is $3 \mathrm{~kg}$.

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Problem 21

Consider a ball falls from some height $H$. Let $e$ be the coclficient ol restitution bolween the ball and the ground and ball rebounds again and again, then find
(a) velocity after $n^{\text {th }}$ strike
(b) height attained af ter $n^{\text {th }}$ strike
(c) total distance travelled by ball before stop
(d) total time of motion.

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Problem 22

$\Lambda$ ball is projected from the ground with the velocily $u$ making an angle $\theta$ with the ground. If the coefficient of restitution is $e$, then find
(a) horizontal range of the ball after $n^{n}$ skrike,
(b) time taken by the ball in between $n^{\text {in }}$ and $(n+1)^{\text {th }}$ surike
(c) total displacement of the ball till it stops
(d) total time of motion.

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Problem 23

'Two blocks of masses $m_{1}$ and $m_{2}$ connected by a light spring rest on a horizontal plane. The coefficient of friction between the masses and the surface is cqual to $\mu$. What minimum constant force $F$ has to be applicd in the horizontal dircction to the block $m$ in order to shift the other block?

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Problem 24

A light spring of spring constant $k$ is kept compressed between two blocks of masses $m_{1}$ and $m_{2}$ on a smooth horizontal surface, When released, the blocks acquire velocitics in opposite dircctions. The spring loses contact with the blocks when it acquires natural length. If the spring was initially compressed through a distance $x$, find the final specds of the two blocks.

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Problem 25

'Iwo blocks of masses $m_{1}$ and $m_{2}$ connected by a light spring of stiffiness $k$ as shown in figure rest on a smooth horizontal plane. Block 2 is shifted a small distance $x$ to the left and then released. Find the velocity of the centre of mass of the system after block 1 breaks off the wall.

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Problem 26

Two blocks 1 and 2 of masses $m$ and $2 m$ respectively are connected by a spring of force constant $k$. The masses are moving to the right with uniform velocity $v$ cach, the heavicr mass, leading the lighter one. 'lhe spring is of natural length in the motion. Block 2 collides head on with a third block 3 of mass $m$, at rest, the collision being completely inclastic. Determine the velocity of blocks at the instant of maximum compression of the spring.

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Problem 27

$\Lambda$ block of mass $m_{1}$ is connected to another block of mass $m_{2}$ by a light spring of spring constant $k$. 'The blocks are kept on a smooth horizontal surface. Initially the blocks are at rest and the spring is unstretched when a constant force $F$ starts acting on the block of mass $m_{2}$ to pull il. Find the maximum clongation of the spring.

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Problem 28

Two blocks of cqual mass $m$ are connected by an unstretched spring and the system is kept at rest on a smooth horizontal surface. A constant force $F$ is applied on one of the blocks pulling it away from the other as shown in figure.
(a) Find the position of the centre of mass at time $t$.
(b) If the elongation of the spring is $x_{n}$ at time $t$, find the displacement of the two blocks at this instant.

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Problem 29

$\Lambda$ disc of radius $r$, and mass $m_{1}$ moving at a speed $v$ undergocs an clastic collision with another stationary disc of radius $r_{2}$ and $m_{2} .$ Find the magnitude and dircction of the velocity of cach after collision in terms o $[$ the impact parameter $d$.

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Problem 30

Two particles $A$ and $B$ of mass $1 \mathrm{~kg}$ and $2 \mathrm{~kg}$ respectively are projected in the directions shown in figure with spoeds $u_{\mathrm{A}}=200 \mathrm{~m} / \mathrm{s}$ and $u_{\mathrm{B}}=50 \mathrm{~m} / \mathrm{s}$. Initially, they were $90 \mathrm{~m}$ apart. Find the maximum height attained by the centre of mass of the particles. $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

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Problem 31

A particle of mass $2 \mathrm{~kg}$ moving with a velocity $5 \hat{i} \mathrm{~m} / \mathrm{s}$ collides head-on with another particle of mass $3 \mathrm{~kg}$ moving with a velocity $-2 \hat{i} \mathrm{~m} / \mathrm{s} .$ \Lambdafter the collision the first particle has speed of $1.6 \mathrm{~m} / \mathrm{s}$ in negative $x$ direction. \Gammaind:
(a) velocity of the cenure of mass after the collision
(b) velocity of the sccond particle after the collision
(c) coefficient of restitution

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Problem 32

(a) $\vec{v}_{\mathrm{CM}}-\frac{m_{1} \vec{u}_{1}+m_{2} \vec{u}_{2}}{m_{1}+m_{2}}-0.8 \hat{i} \mathrm{~m} / \mathrm{s}$
(b) $\vec{v}_{1}--1.6 \hat{i} \mathrm{~m} / \mathrm{s}$
From $\mathrm{c}, \mathrm{m}, m_{1} \vec{u}_{1}+m_{2} \vec{u}_{2}-m_{1} \vec{v}_{1}+m_{2} \vec{v}_{2} \quad \Rightarrow \quad v_{2}-2.4 \hat{i} \mathrm{~m} / \mathrm{s}$
(c) $e-\frac{v_{2}-v_{1}}{u_{1}-u_{2}}-\frac{4}{7}$

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Problem 33

$N$ smooth spheres are placed in a row and their masses arc in the
ratio $1: \frac{1}{e}: \frac{1}{e^{2}} \cdots \frac{1}{e^{n-1}}$ where $e=$ cocflicient of restitution. If a velocity $v$ is given to the first sphere, find the velocity of the last sphere.

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Problem 34

Why is it easier for two men to turn a chop of timber about its centre than to move it translationally?

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Problem 35

'Ihree identical discs $A, B$ and $C$ rest on a smooth horizontal plane. 'lhe disc $\Lambda$ is sel in motion with velocily $v$ along the perpendicular bisector of the lin $B C$ joining the centres of the stationary discs. The distance $B C$ between the centres of the stationary dises $B$ and $C$ is $n$ times the diamcter of cach disc. At what values of $n$ will the disc $A$ recoil, stop, move on afler clastic collision?

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Problem 36

$\Lambda$ particle of mass $m_{1}$ collides clastically with a stationary particle of mass $m_{2}\left(m_{1}<m_{2}\right) .$ Find the maximum angle through which the suriking particle may deviate after the collision.

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Problem 37

A body of mass $M$ with a small block $m$ placed on it rests on a smooth horizontal surfaec. The block is sel in molion in the horizontal direction with a velocity $v$. To what height relative to the initial level will the block rise after breaking off from the body $M ?$ (Friction is absent between $m$ and $M$ ).

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Problem 38

A ball of mass $m$, travelling with velocity $2 \hat{i}+3 \hat{j}$ receives an impulse $-3 m \hat{i}$. What is the velocity of the ball immediately afterwards?

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Problem 39

A particle of mass $2 \mathrm{~kg}$ is initially at rest. A force starts acting on it in one direction whose magnitude changes with time. The force timc graph is shown in figurc.

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