Centre of Mass & Rotational Motion · JEE Main previous year questions

2022 23 questions

1. Q362022 · 29 Jul 2022 · Shift 1

   Two bodies of mass $1\mathrm{kg}$ and $3\mathrm{kg}$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

   1. A.$\hat{i}+2\hat{j}+\hat{k}$
   2. B.$-3\hat{i}-2\hat{j}+\hat{k}$
   3. C.$-2\hat{i}+2\hat{k}$
   4. D.$2\hat{i}-\hat{j}+2\hat{k}$
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   Answer: A

2. Q322022 · 29 Jul 2022 · Shift 2

   Match List I with List II. $$\begin{array}{llll}& \text{List I}& & \text{List II}\\ \text{A.}& \text{Torque}& \text{I.}& \mathrm{Nm}{\mathrm{s}}^{-1}\\ \text{B.}& \text{Stress}& \text{II.}& \mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}\\ \text{C.}& \text{Latent Heat}& \text{III.}& \mathrm{Nm}\\ \text{D.}& \text{Power}& \text{IV.}& \mathrm{N}{\mathrm{m}}^{-2}\end{array}$$ Choose the correct answer from the options given below :

   1. A.A-III, B-II, C-I, D-IV
   2. B.A-III, B-IV, C-II, D-I
   3. C.A-IV, B-I, C-III, D-II
   4. D.A-II, B-III, C-I, D-IV
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   Answer: B

3. Q452022 · 29 Jul 2022 · Shift 2

   The torque of a force $5\hat{i}+3\hat{j}-7\hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2\hat{i}+2\hat{j}+\hat{k}$, then the value of $\tau$ will be

   1. A.$11\hat{i}+19\hat{j}-4\hat{k}$
   2. B.$-11\hat{i}+9\hat{j}-16\hat{k}$
   3. C.$-17\hat{i}+19\hat{j}-4\hat{k}$
   4. D.$17\hat{i}+9\hat{j}+16\hat{k}$
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   Answer: C

4. Q512022 · 29 Jul 2022 · Shift 2· Numerical answer

   A tube of length $50\mathrm{cm}$ is filled completely with an incompressible liquid of mass $250\mathrm{g}$ and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $x\sqrt{F}\mathrm{rad}{\mathrm{s}}^{-1}$. If $F$ be the force exerted by the liquid at the other end then the value of $x$ will be ____.

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   Answer: 4

5. Q572022 · 28 Jul 2022 · Shift 1· Numerical answer
   
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   Answer: 3

6. Q312022 · 27 Jul 2022 · Shift 1

   A torque meter is calibrated to reference standards of mass, length and time each with 5% accuracy. After calibration, the measured torque with this torque meter will have net accuracy of :

   1. A.15%
   2. B.25%
   3. C.75%
   4. D.5%
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   Answer: B

7. Q592022 · 27 Jul 2022 · Shift 1· Numerical answer

   A pulley of radius $1.5\text{ m}$ is rotated about its axis by a force $F=(12t-3t^2)N$ applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5{\text{ kg m}}^2$, the number of rotations made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi }$. The value of K is _________.

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   Answer: 18

8. Q522022 · 26 Jul 2022 · Shift 1· Numerical answer

   A disc of mass $1\mathrm{kg}$ and radius $\mathrm{R}$ is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be $4\sqrt{\frac{x}{3\mathrm{R}}}\mathrm{rad}{\mathrm{s}}^{-1}$ where $x=$ ____. $\left(g=10\mathrm{m}{\mathrm{s}}^{-2}\right)$

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   Answer: 5

9. Q342022 · 25 Jul 2022 · Shift 2
   
   1. A.$\sqrt{2}\text{N},45^{\circ }$
   2. B.$\sqrt{2}\text{N},135^{\circ }$
   3. C.$\frac{2}{\sqrt{3}}\text{N},30^{\circ }$
   4. D.$2\text{N},45^{\circ }$
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   Answer: A

10. Q522022 · 25 Jul 2022 · Shift 2· Numerical answer

    Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $3\text{ m}$ each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt{x}\text{ m}$. The value of x is ___________.

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    Answer: 2

11. Q422022 · 30 Jun 2022 · Shift 1

    Which of the following equations correctly represents a travelling wave having wavelength $\lambda =4.0\text{ cm}$, frequency $\nu =100\text{ Hz}$ and travelling in positive x-axis direction?

    1. A.$y=A\sin [(0.50\pi {\text{ cm}}^{-1})x-(100\pi {\text{ s}}^{-1})t]$
    2. B.$y=A\sin 2\pi [(0.25{\text{ cm}}^{-1})x-(50{\text{ s}}^{-1})t]$
    3. C.$y=A\sin \left[\left(\frac{2\pi }{4}{\text{ cm}}^{-1}\right)x-\left(\frac{2\pi }{100}{\text{ s}}^{-1}\right)t\right]$
    4. D.$y=A\sin \pi [(0.5{\text{ cm}}^{-1})x-(200{\text{ s}}^{-1})t]$
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    Answer: D

12. Q472022 · 30 Jun 2022 · Shift 1

    A hydrogen atom in ground state absorbs $12.09\text{ eV}$ of energy. The orbital angular momentum of the electron is increased by :

    1. A.$1.05\times 10^{-34}\text{ Js}$
    2. B.$2.11\times 10^{-34}\text{ Js}$
    3. C.$3.16\times 10^{-34}\text{ Js}$
    4. D.$4.22\times 10^{-34}\text{ Js}$
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    Answer: B

13. Q522022 · 30 Jun 2022 · Shift 1· Numerical answer
    
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    Answer: 5

14. Q322022 · 28 Jun 2022 · Shift 2

    A ball is spun with angular acceleration $\alpha =6t^2-2t$ where t is in second and $\alpha$ is in $\text{rad}{\text{s}}^{-2}$. At $t=0$, the ball has angular velocity of $10\text{ rad}{\text{s}}^{-1}$ and angular position of $4\text{ rad}$. The most appropriate expression for the angular position of the ball is :

    1. A.$\frac{3}{2}{t}^4-t^2+10t$
    2. B.$\frac{t^4}{2}-\frac{t^3}{3}+10t+4$
    3. C.$\frac{2t^4}{3}-\frac{t^3}{6}+10t+12$
    4. D.$2t^4-\frac{t^3}{2}+5t+4$
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    Answer: B

15. Q352022 · 27 Jun 2022 · Shift 1

    Two blocks of masses 10 kg and 30 kg are placed on the same straight line with coordinates (0, 0) cm and (x, 0) cm respectively. The block of 10 kg is moved on the same line through a distance of 6 cm towards the other block. The distance through which the block of 30 kg must be moved to keep the position of centre of mass of the system unchanged is :

    1. A.4 cm towards the 10 kg block
    2. B.2 cm away from the 10 kg block
    3. C.2 cm towards the 10 kg block
    4. D.4 cm away from the 10 kg block
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    Answer: C

16. Q472022 · 27 Jun 2022 · Shift 1

    A hydrogen atom in its ground state absorbs $10.2\mathrm{eV}$ of energy. The angular momentum of electron of the hydrogen atom will increase by the value of : (Given, Planck's constant $=6.6\times 10^{-34}\mathrm{Js}$).

    1. A.$2.10\times 10^{-34}\mathrm{Js}$
    2. B.$1.05\times 10^{-34}\mathrm{Js}$
    3. C.$3.15\times 10^{-34}\mathrm{Js}$
    4. D.$4.2\times 10^{-34}\mathrm{Js}$
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    Answer: B

17. Q342022 · 27 Jun 2022 · Shift 2

    One end of a massless spring of spring constant $k$ and natural length $l_0$ is fixed while the other end is connected to a small object of mass $m$ lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ about an axis passing through fixed end, then the elongation of the spring will be :

    1. A.$\frac{k-m{\omega }^2{l}_0}{m{\omega }^2}$
    2. B.$\frac{m{\omega }^2{l}_0}{k+m{\omega }^2}$
    3. C.$\frac{m{\omega }^2{l}_0}{k-m{\omega }^2}$
    4. D.$\frac{k+m{\omega }^2{l}_0}{m{\omega }^2}$
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    Answer: C

18. Q352022 · 26 Jun 2022 · Shift 1

    A thin circular ring of mass M and radius R is rotating with a constant angular velocity $2{\text{ rad s}}^{-1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in ${\text{rad s}}^{-1}$).

    1. A.$\frac{M}{(M+m)}$
    2. B.$\frac{(M+2m)}{2M}$
    3. C.$\frac{2M}{(M+2m)}$
    4. D.$\frac{2(M+2m)}{M}$
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    Answer: C

19. Q342022 · 26 Jun 2022 · Shift 2

    A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is

    1. A.$\frac{2}{5}$
    2. B.$\frac{2}{7}$
    3. C.$\frac{1}{5}$
    4. D.$\frac{7}{10}$
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    Answer: B

20. Q342022 · 25 Jun 2022 · Shift 1

    If force $\vec{F}=3\hat{i}+4\hat{j}-2\hat{k}$ acts on a particle position vector $2\hat{i}+\hat{j}+2\hat{k}$ then, the torque about the origin will be :

    1. A.$3\hat{i}+4\hat{j}-2\hat{k}$
    2. B.$-10\hat{i}+10\hat{j}+5\hat{k}$
    3. C.$10\hat{i}+5\hat{j}-10\hat{k}$
    4. D.$10\hat{i}+\hat{j}-5\hat{k}$
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    Answer: B

21. Q332022 · 25 Jun 2022 · Shift 2

    A disc with a flat small bottom beaker placed on it at a distance R from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $\omega$. The coefficient of static friction between the bottom of the beaker and the surface of the disc is $\mu$. The beaker will revolve with the disc if :

    1. A.$R\le \frac{\mu g}{2{\omega }^2}$
    2. B.$R\le \frac{\mu g}{{\omega }^2}$
    3. C.$R\ge \frac{\mu g}{2{\omega }^2}$
    4. D.$R\ge \frac{\mu g}{{\omega }^2}$
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    Answer: B

22. Q542022 · 25 Jun 2022 · Shift 2· Numerical answer

    Moment of Inertia (M.I.) of four bodies having same mass 'M' and radius '2R' are as follows: ${\mathrm{I}}_1=$ M.I. of solid sphere about its diameter ${\mathrm{I}}_2=$ M.I. of solid cylinder about its axis ${\mathrm{I}}_3=$ M.I. of solid circular disc about its diameter ${\mathrm{I}}_4=$ M.I. of thin circular ring about its diameter If $2({\mathrm{I}}_2+{\mathrm{I}}_3)+{\mathrm{I}}_4=x\cdot {\mathrm{I}}_1$, then the value of x will be ____.

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    Answer: 5

23. Q422022 · 24 Jun 2022 · Shift 2

    A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

    1. A.7.5 rad
    2. B.15 rad
    3. C.20 rad
    4. D.30 rad
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    Answer: B