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

2019 34 questions

1. Q322019 · 12 Apr 2019 · Shift 1
   
   1. A.$\sqrt{\frac{a^2+b^2+ab}{2}}$
   2. B.$\sqrt{\frac{a+b}{3}}$
   3. C.$\sqrt{\frac{a^2+b^2+ab}{3}}$
   4. D.$\sqrt{\frac{a+b}{2}}$
   Show answer

   Answer: C

2. Q352019 · 12 Apr 2019 · Shift 1
   
   1. A.
   2. B.
   3. C.
   4. D.
   Show answer

   Answer: B

3. Q352019 · 12 Apr 2019 · Shift 2
   
   1. A.$\frac{\sqrt{3}g}{2r}$
   2. B.$\frac{2g}{(r\sqrt{3})}$
   3. C.$\frac{(g\sqrt{3})}{r}$
   4. D.$\frac{2g}{r}$
   Show answer

   Answer: B

4. Q502019 · 12 Apr 2019 · Shift 2

   A uniform cylindrical rod of length L and radius r, is made from a material whose Young's modulus of Elasticity equals Y. When this rod is heated by temperature T and simultaneously subjected to a net longitudinal compressional force F, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to :

   1. A.$\frac{3F}{(\pi r^{2}YT)}$
   2. B.$\frac{6F}{(\pi r^{2}YT)}$
   3. C.$\frac{F}{(3\pi r^{2}YT)}$
   4. D.$9F(\pi r^{2}YT)$
   Show answer

   Answer: A

5. Q552019 · 12 Apr 2019 · Shift 2
   
   1. A.$\left(\frac{\sqrt{3}}{4}\text{ m},\frac{5}{12}\text{ m}\right)$
   2. B.$\left(\frac{7}{12}\text{ m},\frac{\sqrt{3}}{8}\text{ m}\right)$
   3. C.$\left(\frac{7}{12}\text{ m},\frac{\sqrt{3}}{4}\text{ m}\right)$
   4. D.$\left(\frac{\sqrt{3}}{8}\text{ m},\frac{7}{12}\text{ m}\right)$
   Show answer

   Answer: C

6. Q372019 · 10 Apr 2019 · Shift 1

   A thin disc of mass M and radius R has mass per unit area $\sigma (r)=k{r}^2$ where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

   1. A.$\frac{M{R}^2}{3}$
   2. B.$\frac{M{R}^2}{6}$
   3. C.$\frac{2M{R}^2}{3}$
   4. D.$\frac{M{R}^2}{2}$
   Show answer

   Answer: C

7. Q382019 · 10 Apr 2019 · Shift 1

   Two coaxial discs, having moments of inertia ${\mathrm{I}}_1$ and ${\mathrm{I}}_1/2$, are rotating with respective angular velocities ${\omega }_1$ and ${\omega }_1/2$ , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If ${\mathrm{E}}_{\mathrm{f}}$ and ${\mathrm{E}}_{\mathrm{i}}$ are the final and initial total energies, then $({\mathrm{E}}_{\mathrm{f}}-{\mathrm{E}}_{\mathrm{i}})$ is:

   1. A.$\frac{I_1{\omega }_1^2}{24}$
   2. B.$\frac{I_1{\omega }_1^2}{12}$
   3. C.$\frac{3}{8}{I}_1{\omega }_1^2$
   4. D.$\frac{I_1{\omega }_1^2}{6}$
   Show answer

   Answer: A

8. Q392019 · 10 Apr 2019 · Shift 1

   A particle of mass m is moving along a trajectory given by $x=x_0+a\cos {\omega }_{1}t$ $y=y_0+b\sin {\omega }_{2}t$ The torque, acting on the particle about the origin, at t = 0 is :

   1. A.Zero
   2. B.$+m{y}_{0}a{\omega }_1^2\hat{k}$
   3. C.$-m\left(x_{0}b{\omega }_2^2-y_{0}a{\omega }_1^2\right)\hat{k}$
   4. D.$m\left(-x_{0}b+y_{0}a\right){\omega }_1^2\hat{k}$
   Show answer

   Answer: B

9. Q362019 · 10 Apr 2019 · Shift 2
   
   1. A.$7.9\times 10^{-6}\mathrm{Nm}$
   2. B.$4.0\times 10^{-6}\mathrm{Nm}$
   3. C.$2.0\times 10^{-5}\mathrm{Nm}$
   4. D.$1.6\times 10^{-5}\mathrm{Nm}$
   Show answer

   Answer: C

10. Q462019 · 10 Apr 2019 · Shift 2

    A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of $\frac{7M}{8}$ and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let ${\mathrm{I}}_1$ be the moment of inertia of the disc about its axis and ${\mathrm{I}}_2$ be the moment of inertia of the new sphere about its axis. The ratio ${\mathrm{I}}_1/{\mathrm{I}}_2$ is given by :

    1. A.65
    2. B.140
    3. C.185
    4. D.285
    Show answer

    Answer: B

11. Q502019 · 10 Apr 2019 · Shift 2

    The time dependence of the position of a particle of mass m = 2 is given by $\vec{r}(t)=2t\hat{i}-3t^2\hat{j}$. Its angular momentum, with respect to the origin, at time t = 2 is

    1. A.$36\hat{k}$
    2. B.- 48 $\hat{k}$
    3. C.$-34\left(\hat{k}-\hat{i}\right)$
    4. D.$48\left(\hat{i}+\hat{j}\right)$
    Show answer

    Answer: B

12. Q582019 · 10 Apr 2019 · Shift 2

    The elastic limit of brass is 379 MPa. What should be the minimum diameter of a brass rod if it is to support a 400 N load without exceeding its elastic limit?

    1. A.1.16 mm
    2. B.1.36 mm
    3. C.1.00 mm
    4. D.0.90 mm
    Show answer

    Answer: A

13. Q322019 · 9 Apr 2019 · Shift 1

    An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is $\bar{v}$, m is its mass and $k_B$ is Boltzmann constant, then its temperature will be :

    1. A.$\frac{m{\bar{v}}^2}{5k_B}$
    2. B.$\frac{m{\bar{v}}^2}{6k_B}$
    3. C.$\frac{m{\bar{v}}^2}{7k_B}$
    4. D.$\frac{m{\bar{v}}^2}{3k_B}$
    Show answer

    Answer: D

14. Q502019 · 9 Apr 2019 · Shift 1

    The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane. : (i) a ring of radius R, (ii) a solid cylinder of radius R/2 and (iii) a solid sphere of radius R/4 . If in each case, the speed of the centre of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is :

    1. A.20 : 15 : 14
    2. B.4 : 3 : 2
    3. C.2 : 3 : 4
    4. D.10 : 15 : 7
    Show answer

    Answer: A

15. Q512019 · 9 Apr 2019 · Shift 1

    A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$, where $\theta$ is the angle by which it has rotated, is given as $k{\theta }^2$. If its moment of inertia is I then the angular acceleration of the disc is :

    1. A.$\frac{k}{4I}\theta$
    2. B.$\frac{k}{I}\theta$
    3. C.$\frac{k}{2I}\theta$
    4. D.$\frac{2k}{I}\theta$
    Show answer

    Answer: D

16. Q562019 · 9 Apr 2019 · Shift 1

    A rectangular coil (Dimension 5 cm × 2.5 cm) with 100 turns, carrying a current of 3 A in the clock- wise direction is kept centered at the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through 45° about Z-axis, then the torque on the coil is :

    1. A.0.42 Nm
    2. B.0.55 Nm
    3. C.0.38 Nm
    4. D.0.27 Nm
    Show answer

    Answer: D

17. Q422019 · 9 Apr 2019 · Shift 2

    Moment of inertia of a body about a given axis is $1.5\mathrm{kg}{\mathrm{m}}^2$. Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular accleration of $20rad/{\mathrm{s}}^2$ must be applied about the axis for a duration of :-

    1. A.2.5 s
    2. B.3 s
    3. C.5s
    4. D.2 s
    Show answer

    Answer: D

18. Q552019 · 9 Apr 2019 · Shift 2

    A thin smooth rod of length L and mass M is rotating freely with angular speed ${\omega }_0$ about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system , when the beads reach the opposite ends of the rod, will be :-

    1. A.$\frac{M{\omega }_0}{M+3m}$
    2. B.$\frac{M{\omega }_0}{M+m}$
    3. C.$\frac{M{\omega }_0}{M+6m}$
    4. D.$\frac{M{\omega }_0}{M+2m}$
    Show answer

    Answer: C

19. Q462019 · 8 Apr 2019 · Shift 2

    A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion HBGO is cut-off, the coordinates of the centre of mass of the remaining portion will be :- [Figure: rectangle with D(0, 0), C(a, 0), B(a, b), A(0, b); H and F are midpoints of AB and DC, E and G are midpoints of AD and BC; O is the centre $\left(\frac{a}{2},\frac{b}{2}\right)$; the shaded portion HBGO is the top-right quarter]

    1. A.$\left(\frac{5a}{3},\frac{5b}{3}\right)$
    2. B.$\left(\frac{2a}{3},\frac{2b}{3}\right)$
    3. C.$\left(\frac{5a}{12},\frac{5b}{12}\right)$
    4. D.$\left(\frac{3a}{4},\frac{3b}{4}\right)$
    Show answer

    Answer: C

20. Q492019 · 8 Apr 2019 · Shift 2

    A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time t = 0.01s, remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to :-

    1. A.0.28
    2. B.0.02
    3. C.0.3
    4. D.0.5
    Show answer

    Answer: D

21. Q382019 · 12 Jan 2019 · Shift 1

    Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

    1. A.16 cm
    2. B.12 cm
    3. C.14 cm
    4. D.18 cm
    Show answer

    Answer: A

22. Q402019 · 12 Jan 2019 · Shift 1
    
    1. A.${\vec{r}}_{cm}=\frac{11}{8}L\hat{x}+\frac{8}{8}L\hat{y}$
    2. B.${\vec{r}}_{cm}=\frac{5}{8}L\hat{x}+\frac{13}{8}L\hat{y}$
    3. C.${\vec{r}}_{cm}=\frac{13}{8}L\hat{x}+\frac{5}{8}L\hat{y}$
    4. D.${\vec{r}}_{cm}=\frac{3}{8}L\hat{x}+\frac{11}{8}L\hat{y}$
    Show answer

    Answer: C

23. Q562019 · 12 Jan 2019 · Shift 1
    
    1. A.$\frac{1}{2\pi }\sqrt{\frac{3k}{m}}$
    2. B.$\frac{1}{2\pi }\sqrt{\frac{6k}{m}}$
    3. C.$\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
    4. D.$\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$
    Show answer

    Answer: B

24. Q322019 · 12 Jan 2019 · Shift 2
    
    1. A.$6\mathrm{kg}\cdot {\mathrm{m}}^2/s$
    2. B.$8\mathrm{kg}\cdot {\mathrm{m}}^2/s$
    3. C.$2\mathrm{kg}\cdot {\mathrm{m}}^2/s$
    4. D.$3\mathrm{kg}\cdot {\mathrm{m}}^2/s$
    Show answer

    Answer: A

25. Q392019 · 12 Jan 2019 · Shift 2
    
    1. A.
    2. B.
    3. C.
    4. D.
    Show answer

    Answer: C