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

1. A.go straight
2. B.turn left and right alternately
3. C.turn left
4. D.turn right

1. A.$\vec{L}=mv\left[\frac{R}{\sqrt{2}}+a\right]\hat{k}$ when the particle is moving from B to C.
2. B.$\vec{L}=\frac{mv}{\sqrt{2}}R\hat{k}$ when the particle is moving from D to A.
3. C.$\vec{L}=-\frac{mv}{\sqrt{2}}R\hat{k}$ when the particle is moving from A to B
4. D.$\vec{L}=mv\left[\frac{R}{\sqrt{2}}-a\right]\hat{k}$ when the particle is moving from C to D.

Distance of the center 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 :

1. A.$\frac{5h}{8}$
2. B.$\frac{3h^2}{8R}$
3. C.$\frac{h^2}{4R}$
4. D.$\frac{3h}{4}$

From a solid sphere of mass $M$ and radius $R$ a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its face is:

1. A.$\frac{4M{R}^2}{9\sqrt{3}\pi }$
2. B.$\frac{4M{R}^2}{3\sqrt{3}\pi }$
3. C.$\frac{M{R}^2}{32\sqrt{2}\pi }$
4. D.$\frac{M{R}^2}{16\sqrt{2}\pi }$

Match List - I (Fundamental Experiment) with List - II (its conclusion) and select the correct option from the choices given below the list: | List - I | List - II | | :--- | :--- | | A. Franck-Hertz | (i) Particle nature of light | | B. Photo-electric | (ii) Discrete energy levels of atom | | C. Davison-Germer experiment | (iii) Wave nature of electron | | | (iv) Structure of atom |

1. A.$A-ii;B-i,C-iii$
2. B.$A-iv;B-iii,C-ii$
3. C.$A-i;B-iv,C-iii$
4. D.$A-ii;B-iv,C-iii$

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

1. A.A screw gauge having 100 divisions in the circular scale and pitch as 1 mm.
2. B.A screw gauge having 50 divisions in the circular scale and pitch as 1 mm.
3. C.A meter scale.
4. D.A vernier calliper where the 10 divisions in vernier scale matches with 9 divisions in main scale and main scale has 10 divisions in 1 cm.

Three rods of Copper, Brass and Steel are welded together to form a $Y$ shaped structure. Area of cross - section of each rod $=4\mathrm{c}{\mathrm{m}}^2$. End of copper rod is maintained at $100^{\circ }\mathrm{C}$ where as ends of brass and steel are kept at $0^{\circ }\mathrm{C}$. Lengths of the copper, brass and steel rods are $46$, $13$ and $12\mathrm{cms}$ respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are $0.92,0.26$ and $0.12\mathrm{CGS}$ units respectively. Rate of heat flow through copper rod is:

1. A.$1.2cal/s$
2. B.$2.4cal/s$
3. C.$4.8cal/s$
4. D.$6.0cal/s$

1. A.$1.57\mathrm{W}$
2. B.$2.97\mathrm{W}$
3. C.$14.85\mathrm{W}$
4. D.$29.7\mathrm{W}$

A thick-walled hollow sphere has outside radius $R_0$. It rolls down an incline without slipping and its speed at the bottom is $v_0$. Now the incline is waxed, so that it is practically frictionless and the sphere is observed to slide down (without any rolling). Its speed at the bottom is observed to be $5v_0/4$. The radius of gyration of the hollow sphere about an axis through its centre is

1. A.$3R_0/2$
2. B.$3R_0/4$
3. C.$9R_0/16$
4. D.$3R_0$

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

This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: When moment of inertia $I$ of a body rotating about an axis with angular speed $\omega$ increases, its angular momentum $L$ is unchanged but the kinetic energy $K$ increases if there is no torque applied on it. Statement 2: $L=I\omega$, kinetic energy of rotation $=\frac{1}{2}I{\omega }^2$

1. A.Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1.
2. B.Statement 1 is false, Statement 2 is true.
3. C.Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of the Statement 1.
4. D.Statement 1 is true, Statement 2 is false.

1. A.$v_A=v_B=0.5m/s$
2. B.$v_A=0.5m/s<v_B$
3. C.$v_A=0.5m/s>v_B$
4. D.$v_A=v_B=2m/s$

A circular hole of diameter $R$ is cut from a disc of mass $M$ and radius $R$; the circumference of the cut passes through the centre of the disc. The moment of inertia of the remaining portion of the disc about an axis perpendicular to the disc and passing through its centre is

1. A.$\left(\frac{15}{32}\right)M{R}^2$
2. B.$\left(\frac{1}{8}\right)M{R}^2$
3. C.$\left(\frac{3}{8}\right)M{R}^2$
4. D.$\left(\frac{13}{32}\right)M{R}^2$

For a particle in uniform circular motion the acceleration $\vec{a}$ at a point P(R, $\theta$) on the circle of radius R is (here $\theta$ is measured from the x-axis)

1. A.$-\frac{v^2}{R}\cos \theta \hat{i}+\frac{v^2}{R}\sin \theta \hat{j}$
2. B.$-\frac{v^2}{R}\sin \theta \hat{i}+\frac{v^2}{R}\cos \theta \hat{j}$
3. C.$-\frac{v^2}{R}\cos \theta \hat{i}-\frac{v^2}{R}\sin \theta \hat{j}$
4. D.$\frac{v^2}{R}\hat{i}+\frac{v^2}{R}\hat{j}$

1. A.$-mg{v}_0{t}^2\cos \theta \hat{j}$
2. B.$mg{v}_{0}t\cos \theta \hat{k}$
3. C.$-\frac{1}{2}mg{v}_0{t}^2\cos \theta \hat{k}$
4. D.$\frac{1}{2}mg{v}_0{t}^2\cos \theta \hat{i}$

A thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its center of mass rises to a maximum height of:

1. A.$\frac{1}{6}\frac{l\omega }{g}$
2. B.$\frac{1}{2}\frac{l^2{\omega }^2}{g}$
3. C.$\frac{1}{6}\frac{l^2{\omega }^2}{g}$
4. D.$\frac{1}{3}\frac{l^2{\omega }^2}{g}$

1. A.${\sin }^{-1}\left(\sqrt{3}/2\right)$
2. B.${\sin }^{-1}\left(2/\sqrt{3}\right)$
3. C.${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
4. D.${\sin }^{-1}\left(1/2\right)$

Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the center of mass through a distance $d$, by what distance should the second particle is moved, so as to keep the center of mass at the same position?

1. A.$\frac{m_2}{m_1}d$
2. B.$\frac{m_1}{m_1+m_2}d$
3. C.$\frac{m_1}{m_2}d$
4. D.$d$

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $l$. The moment of inertia of this system about an axis passing through $A$ and parallel to $BD$ is

1. A.$2m{l}^2$
2. B.$\sqrt{3}m{l}^2$
3. C.$3m{l}^2$
4. D.$m{l}^2$

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity ${\omega }^{\prime }=$

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

1. A.$F(\hat{i}-\hat{j})$
2. B.$-F(\hat{i}+\hat{j})$
3. C.$F(\hat{i}+\hat{j})$
4. D.$-F(\hat{i}-\hat{j})$

An annular ring with inner and outer radii $R_1$ and $R_2$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, $\frac{F_1}{F_2}$ is

1. A.${\left(\frac{R_1}{R_2}\right)}^2$
2. B.$\frac{R_2}{R_1}$
3. C.$\frac{R_1}{R_2}$
4. D.$1$

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 center of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

1. A.does not shift
2. B.depends on height of breaking
3. C.body $B$
4. D.body $C$

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the center is

1. A.$\frac{2}{5}M{r}^2$
2. B.$\frac{1}{4}Mr$
3. C.$\frac{1}{2}M{r}^2$
4. D.$M{r}^2$

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected ?

1. A.Angular velocity
2. B.Angular momentum
3. C.Moment of inertia
4. D.Rotational kinetic energy