Rotational Motion
270 Questions
Start Allen Test
JEE (Main) 2023
Q1
Allen
Rotational Motion
Numerical
A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\frac{2}{x}}$ . The value of x is ____.
Correct Answer: 5
Explanation:
5
JEE (Main) 2023
Q2
Allen
Rotational Motion
Numerical
For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}$ . The value of x is ____.
Correct Answer: 2
Explanation:
2
JEE (Main) 2023
Q3
Allen
Rotational Motion
Numerical
A uniform solid cylinder with radius R and length L has moment of inertia $I_{1}$ , about the axis of cylinder. A concentric solid cylinder of radius $R'=\frac{R}{2}$ and length $L'=\frac{L}{2}$ is caned out of the original cylinder. If $I_{2}$ is the moment of inertia of the carved-out portion to the cylinder then $\frac{I_{1}}{I_{2}}=$ ____. (Both $I_{1}$ and $I_{2}$ are about the axis of the cylinder)
Correct Answer: 32
Explanation:
32
JEE (Advanced) 2023
Q4
Allen
JEE Advanced PYQ
MCQ
A bar of mas $M = 1.00 \, kg$ and length $L = 0.20 \, m$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m = 0.10 \, kg$ is moving on the same horizontal surface with $5.00 \, ms^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$ . After this elastic collision, the bar rotates with an angular velocity $\omega$ . Which of the following statement is correct?
A.
$\omega = 6.98\mathrm{rad}s^{-1}$ and $v = 4.30ms^{-1}$
B.
$\omega = 3.75$ rad $s^{-1}$ and $v = 4.30$ ms $^{-1}$
C.
$\omega = 3.75$ rad $s^{-1}$ and $v = 10.0$ ms $^{-1}$
D.
$\omega = 6.80$ rad $s^{-1}$ and $v = 4.10$ ms $^{-1}$
JEE (Advanced) 2023
Q5
Allen
JEE Advanced PYQ
Numerical
Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{-8}$ N m rad $^{-1}$ . The angular frequency of the oscillations is $n \times 10^{-3}$ rad s $^{-1}$ . The value of n is ____.
Correct Answer: 10
Explanation:
Ans. (10)
$T = \frac {2 \pi}{\omega} = 2 \pi \sqrt {\frac {I}{C}}$
$\Rightarrow \omega = \sqrt {\frac {C}{I}}$
Where I = moment of inertia
$I = (3 0) (4) ^ {2} + (2 0) (6) ^ {2}$
$= 1 2 0 0 g m \cdot c m ^ {2}$
$= 1. 2 \times 1 0 ^ {- 4} k g \cdot m ^ {2}$
$\Rightarrow \omega = \sqrt {\frac {1 . 2 \times 1 0 ^ {- 8}}{1 . 2 \times 1 0 ^ {- 4}}}$
$\Rightarrow \omega = \sqrt {1 0 ^ {- 4}}$
$\omega = (1 0 ^ {- 2})$
$n \times 1 0 ^ {- 3} = 1 0 ^ {- 2} \Rightarrow n = 1 0$
$T = \frac {2 \pi}{\omega} = 2 \pi \sqrt {\frac {I}{C}}$
$\Rightarrow \omega = \sqrt {\frac {C}{I}}$
Where I = moment of inertia
$I = (3 0) (4) ^ {2} + (2 0) (6) ^ {2}$
$= 1 2 0 0 g m \cdot c m ^ {2}$
$= 1. 2 \times 1 0 ^ {- 4} k g \cdot m ^ {2}$
$\Rightarrow \omega = \sqrt {\frac {1 . 2 \times 1 0 ^ {- 8}}{1 . 2 \times 1 0 ^ {- 4}}}$
$\Rightarrow \omega = \sqrt {1 0 ^ {- 4}}$
$\omega = (1 0 ^ {- 2})$
$n \times 1 0 ^ {- 3} = 1 0 ^ {- 2} \Rightarrow n = 1 0$
JEE (Advanced) 2023
Q6
Allen
JEE Advanced PYQ
MSQ
An annular disk of mass M, inner radius a and outer radius b is placed on a horizontal surface with coefficient of friction $\mu$ , as shown in the figure. At some time, an impulse $J_{0}\hat{x}$ is applied at a height h above the center of the disk. If $h = h_{m}$ then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?
A.
For $\mu \neq 0$ and $a \to 0$ , $h_m = b / 2$
B.
For $\mu \neq 0$ and $a \to b$ , $h_m = b$
C.
For $h = h_{m}$ , the initial angular velocity does not depend on the inner radius a.
D.
For $\mu = 0$ and h = 0, the wheel always slides without rolling.
JEE (Advanced) 2023
Q7
Allen
JEE Advanced PYQ
Numerical
A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal xy-plane. The coin is tossed vertically up (+z direction) by applying an impulse of $\sqrt{\frac{\pi}{2}} \times 10^{-2} N - s$ at a distance 2/3 cm from its center. The coin spins about its diameter and moves along the +z direction. By the time the coin reaches back to its initial position, it completes n rotations. The value of n is _. [Given: The acceleration due to gravity g = 10 ms $^{-2}$ ]
Correct Answer: 30
Explanation:
Ans. (30)
J = m v
$J r = I _ {\mathrm{c}} \omega$
... (1)
$J _ {C} = \frac {1}{4} m R ^ {2}$
... (2)
... (3)
$t = \frac {2 v}{g}$
... (4)
$\theta = 2 \pi N = \omega t$
... (5) $\therefore N = 30$
## JEE (Main) Practice Paper
J = m v
$J r = I _ {\mathrm{c}} \omega$
... (1)
$J _ {C} = \frac {1}{4} m R ^ {2}$
... (2)
... (3)
$t = \frac {2 v}{g}$
... (4)
$\theta = 2 \pi N = \omega t$
... (5) $\therefore N = 30$
## JEE (Main) Practice Paper
JEE (Main) 2022
Q8
Allen
Rotational Motion
Numerical
A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be $x \times 10^{-2}$ kg. The value of x is
Correct Answer: 2
Explanation:
Ans. (2)
Say the distance of central line from instantaneous axis of rotation is r. Then r from the point on left becomes lesser than that for right.
So, $v_{left point} = \omega r' < \omega r = v_{right point}$ .
So, the roller will turn to left.
Say the distance of central line from instantaneous axis of rotation is r. Then r from the point on left becomes lesser than that for right.
So, $v_{left point} = \omega r' < \omega r = v_{right point}$ .
So, the roller will turn to left.
JEE (Advanced) 2022
Q9
Allen
JEE Advanced PYQ
Numerical
At time t = 0, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha = \frac{2}{3}$ rad s $^{-2}$ . A small stone is stuck to the disk. At t = 0, it is at the contact point of the disk and the plane. Later, at time $t = \sqrt{\pi} s$ , the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is $\frac{1}{2} + \frac{x}{10}$ . The value of x is ____. [Take g = 10 m s $^{-2}$ .]
Correct Answer: 0.52
Explanation:
Ans. (3) $\frac{mR^2}{2}\omega = \left(\frac{mR^2}{2} + mR^2\right)\omega'$ $\therefore \omega' = \frac{\omega}{3}$
JEE (Advanced) 2022
Q10
Allen
JEE Advanced PYQ
Numerical
A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination $\theta = 30^{\circ}$ from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance r = 0.5 m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ____ $ms^{-2}$ . (Take $g = 10 \, m s^{-2}$ .)
Correct Answer: 2.85 to 2.86
Explanation:
Ans. (A-R), (B-S), (C-P), (D-Q)
(A) $V_{cm} = l$ and $\omega R = 1$ (B) $v_{rel\ 1/3} \neq 0$ ∴ Not a rigid body
(C) $v_{2} = \hat{i}, v_{3} = 0$ i.e. rotating at same position only (p).
(D) $V_{i} = -\hat{i} - \hat{j}$ $V_{4} = 0$ (Lowest point) so (Q).
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
(A) $V_{cm} = l$ and $\omega R = 1$ (B) $v_{rel\ 1/3} \neq 0$ ∴ Not a rigid body
(C) $v_{2} = \hat{i}, v_{3} = 0$ i.e. rotating at same position only (p).
(D) $V_{i} = -\hat{i} - \hat{j}$ $V_{4} = 0$ (Lowest point) so (Q).
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
## Important Notes
The image contains only background horizontal lines, which are stylistic or background elements and not text placeholders. According to the correction guidelines, such lines must be ignored. Therefore, the correct OCR output is empty.
(no text)
JEE (Advanced) 2022
Q11
Allen
JEE Advanced PYQ
Numerical
A particle of mass 1 kg is subjected to a force which depends on the position as $\vec{F} = -k(x\hat{i} + y\hat{j})kgms^{-2}$ with $k = 1 kgs^{-2}$ . At time $t = 0$ , the particle's position $\vec{r} = \left(\frac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{j}\right)m$ and its velocity $\vec{v} = \left(-\sqrt{2}\hat{i} + \sqrt{2}\hat{j} + \frac{2}{\pi}\hat{k}\right)ms^{-1}$ . Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z = 0.5$ m, the value of $(x v_y - y v_x)$ is ____ $m^2 s^{-1}$ .
Correct Answer: 3
Explanation:
Ans. (3)
Torque about origin is zero
So angular momentum about origin remains conserved.
$\left| \begin{array}{c c c} \hat {i} & \hat {j} & \hat {k} \\ \frac {1}{\sqrt {2}} & \sqrt {2} & 0 \\ - \sqrt {2} & \sqrt {2} & \frac {2}{\pi} \end{array} \right| = \left| \begin{array}{c c c} \hat {i} & \hat {j} & \hat {k} \\ x & y & 0. 5 \\ v _ {x} & v _ {y} & \frac {2}{\pi} \end{array} \right|$
$\hat {\imath} \left[ \sqrt {2} \times \frac {2}{\pi} \right] - \hat {\jmath} \left[ \frac {\sqrt {2}}{\pi} \right] + \hat {k} [ 1 + 2 ]$
$= \hat {\imath} \left[ \frac {y \times 2}{\pi} - 0. 5 v _ {y} \right] - \hat {\jmath} \left[ \frac {x \times 2}{\pi} - 0. 5 v _ {x} \right] + \hat {k} \big [ x v _ {y} - y v _ {x} \big ]$
$x _ {v y} - y _ {v x} = 3$
Torque about origin is zero
So angular momentum about origin remains conserved.
$\left| \begin{array}{c c c} \hat {i} & \hat {j} & \hat {k} \\ \frac {1}{\sqrt {2}} & \sqrt {2} & 0 \\ - \sqrt {2} & \sqrt {2} & \frac {2}{\pi} \end{array} \right| = \left| \begin{array}{c c c} \hat {i} & \hat {j} & \hat {k} \\ x & y & 0. 5 \\ v _ {x} & v _ {y} & \frac {2}{\pi} \end{array} \right|$
$\hat {\imath} \left[ \sqrt {2} \times \frac {2}{\pi} \right] - \hat {\jmath} \left[ \frac {\sqrt {2}}{\pi} \right] + \hat {k} [ 1 + 2 ]$
$= \hat {\imath} \left[ \frac {y \times 2}{\pi} - 0. 5 v _ {y} \right] - \hat {\jmath} \left[ \frac {x \times 2}{\pi} - 0. 5 v _ {x} \right] + \hat {k} \big [ x v _ {y} - y v _ {x} \big ]$
$x _ {v y} - y _ {v x} = 3$
JEE (Advanced) 2022
Q12
Allen
JEE Advanced PYQ
MCQ
A flat surface of a thin uniform disk A of radius R is glued to a horizontal table. Another thin uniform disk B of mass M and with the same radius R rolls without slipping on the circumference of A, as shown in the figure. A flat surface of B also lies on the plane of the table. The center of mass of B has fixed angular speed $\omega$ about the vertical axis passing through the center of A. The angular momentum of B is $nM\omega R^{2}$ with respect to the center of A. Which of the following is the value of n?
A.
2
B.
5
C.
$\frac{7}{2}$
D.
$\frac{9}{2}$
JEE (Main) 2021
Q13
Allen
JEE Main PYQ
MCQ
A triangular plate is shown. A force $\vec{F} = 4\hat{i} - 3\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are :
A.
$- 15 - 20 \sqrt{3}$ , $15 - 20 \sqrt{3}$
B.
$15 + 20\sqrt{3}, 15 - 20\sqrt{3}$
C.
$15 - 20\sqrt{3}, 15 + 20\sqrt{3}$
D.
$-15 + 20\sqrt{3}, 15 + 20\sqrt{3}$
JEE (Main) 2021
Q14
Allen
Rotational Motion
MCQ
A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:
[The coefficient of static friction, $\mu_{s}$ , is 0.4]
A.
$\frac{7}{2}mg$
B.
5 mg
C.
$\frac{mg}{5}$
D.
0
JEE (Main) 2021
Q15
Allen
Rotational Motion
Numerical
A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass 'm' travelling along the surface hits at one end of the rod with a velocity 'u' in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses $\left(\frac{m}{M}\right)$ is $\frac{1}{x}$ . The value of 'x' will be ____.
Correct Answer: 4
Explanation:
Ans. (4)
Different
Different
JEE (Main) 2021
Q16
Allen
Rotational Motion
Numerical
In the given figure, two wheels P and Q are connected by a belt B. The radius of P is three times as that of Q. In case of same rotational kinetic energy, the ratio of rotational inertias $\left(\frac{I_{1}}{I_{2}}\right)$ will be x:1. The value of x will be ____.
Correct Answer: 4
Explanation:
Ans. (4)
$\begin{array}{r l} & {\vec {r _ {c}} = \frac {1 \times 1 (\cos 6 0 \hat {i} + \sin 6 0 \hat {j}) + 3 \hat {i}}{6}} \\ & {\quad = \frac {3 . 5 \hat {i} + \frac {\sqrt {3}}{2} \hat {j}}{6}} \end{array}$
$I = I _ {\mathrm{C}} + m r _ {\mathrm{C}} ^ {2} = 1 \times 1 2 + 3 \times 1 2$
$\begin{array}{r l} I _ {\mathrm{C}} & = 4 - \frac {6 \times \left(\frac {4 9}{4} + \frac {3}{4}\right)}{6 ^ {2}} \\ & = 4 - \frac {5 2}{2 4} \\ & = \frac {2 1}{6} k g - m ^ {2} = 3. 5 k g \cdot m ^ {2} \end{array}$
$\begin{array}{r l} & {\vec {r _ {c}} = \frac {1 \times 1 (\cos 6 0 \hat {i} + \sin 6 0 \hat {j}) + 3 \hat {i}}{6}} \\ & {\quad = \frac {3 . 5 \hat {i} + \frac {\sqrt {3}}{2} \hat {j}}{6}} \end{array}$
$I = I _ {\mathrm{C}} + m r _ {\mathrm{C}} ^ {2} = 1 \times 1 2 + 3 \times 1 2$
$\begin{array}{r l} I _ {\mathrm{C}} & = 4 - \frac {6 \times \left(\frac {4 9}{4} + \frac {3}{4}\right)}{6 ^ {2}} \\ & = 4 - \frac {5 2}{2 4} \\ & = \frac {2 1}{6} k g - m ^ {2} = 3. 5 k g \cdot m ^ {2} \end{array}$
JEE (Main) 2021
Q17
Allen
Rotational Motion
MCQ
Two discs have moments of inertia $I_{1}$ and $I_{2}$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega_{1}$ and $\omega_{2}$ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:
A.
$\frac{I_{1}I_{2}}{(I_{1}+I_{2})}\left(\omega_{1}-\omega_{2}\right)^{2}$
B.
$\frac{\left(I_{1}-I_{2}\right)^{2}\omega_{1}\omega_{2}}{2\left(I_{1}+I_{2}\right)}$
C.
$\frac{I_{1}I_{2}}{2\left(I_{1}+I_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$
D.
$\frac{\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$
JEE (Main) 2021
Q18
Allen
Rotational Motion
MCQ
The solid cylinder of length 80 cm and mass M has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is 2.7 kg m $^{2}$ .
A.
$14.9 \, kg / m^{3}$
B.
$7.5 \times 10^{1} \, kg / m^{3}$
C.
$7.5 \times 10^{2} \, kg/m^{3}$
D.
$1.49 \times 10^{2} \, kg / m^{3}$
JEE (Main) 2021
Q19
Allen
Rotational Motion
Numerical
Consider a badminton racket with length scales as shown in the figure.
If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, $\frac{r}{2}$ distance from the end A of the handle will be ..... $Mr^{2}$ .
If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, $\frac{r}{2}$ distance from the end A of the handle will be ..... $Mr^{2}$ .
Correct Answer: 1
Explanation:
Ans. (1)
Let mass and side of cube be $M'$ and $\sqrt{3}$ a = 2R
$M ^ {\prime} = \frac {M}{\frac {4}{3} \pi R ^ {3}} a ^ {3}$
Moment of Inertia of cube = $\frac{M'a^{2}}{6}$
$= \left(\frac {M}{\frac {4}{3} \pi R ^ {3}} a ^ {3}\right) \frac {a ^ {2}}{6} = \boxed {\frac {4 M R ^ {2}}{9 \sqrt {3} \pi}}$
Let mass and side of cube be $M'$ and $\sqrt{3}$ a = 2R
$M ^ {\prime} = \frac {M}{\frac {4}{3} \pi R ^ {3}} a ^ {3}$
Moment of Inertia of cube = $\frac{M'a^{2}}{6}$
$= \left(\frac {M}{\frac {4}{3} \pi R ^ {3}} a ^ {3}\right) \frac {a ^ {2}}{6} = \boxed {\frac {4 M R ^ {2}}{9 \sqrt {3} \pi}}$
JEE (Advanced) 2021
Q20
Allen
JEE Advanced PYQ
MSQ
A horizontal force F is applied at the center of mass of a cylindrical object of mass m and radius R, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$ . The center of mass of the object has an acceleration a. The acceleration due to gravity is g. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?
A.
For the same F, the value of a does not depend on whether the cylinder is solid or hollow.
B.
For a solid cylinder, the maximum possible value of $a$ is $2\mu g$ .
C.
The magnitude of the frictional force on the object due to the ground is always $\mu mg$ .
D.
For a thin-walled hollow cylinder, $a = \frac{F}{2m}$ .
JEE (Advanced) 2021
Q21
Allen
JEE Advanced PYQ
MSQ
A particle of mass M = 0.2 kg is initially at rest in the xy-plane at a point $(x = -l, y = -h)$ , where l = 10 m and h = 1m. The particle is accelerated at time t = 0 with a constant acceleration $a = 10 \, m/s^{2}$ along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{\tau}$ , respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive x, y and z-directions, respectively. If $\hat{k} = \hat{i} \times \hat{j}$ then which of the following statement(s) is(are) correct?
A.
The particle arrives at the point $(x = l, y = -h)$ at time t = 2s
B.
$\vec{\tau} = 2\hat{k}$ when the particle passes through the point (x = l, y = -h)
C.
$\vec{L} = 4\hat{k}$ when the particle passes through the point $(x = l, y = -h)$
D.
$\vec{\tau} = \hat{k}$ when the particle passes through the point $(x = 0, y = -h)$
JEE (Advanced) 2021
Q22
Allen
JEE Advanced PYQ
Numerical
A thin rod of mass M and length a is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius a/4 is pivoted on this rod with its center at a distance a/4 from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$ . The total angular momentum of the system about the point O is $\left(\frac{Ma^{2}\Omega}{48}\right)n$ . The value of n is ____.
Correct Answer: 49
Explanation:
Ans. (5)
$4 0 - T _ {A} = 4 a\tag{... (i}$
$T _ {B} - 2 0 = 2 a$
$(T _ {A} - T _ {B}) \times 0. 1 \frac {M (0 . 1) ^ {2}}{2} \alpha\tag{... (ii}$
... (iii)
$T _ {A} - T _ {B} = 2 \alpha\tag{... (vi}$
$a = \alpha R \Rightarrow a = \frac {\alpha}{1 0}\tag{... (iv}$
$\text { Now } (1) + (2) \Rightarrow T _ {B} - T _ {A} + 2 0 = 6 a$
$\therefore - 2 \alpha + 2 0 = 6 a$
$\therefore \quad - 2 \alpha + 2 0 = \frac {6 \alpha}{1 0} \Rightarrow 2 0 \frac {2 6 \alpha}{1 0}$
$\therefore \alpha = \frac {2 0 0}{2 6}$
$\therefore \alpha = \frac {1 0 0}{1 3}$
$\therefore \quad a = \frac {\alpha}{1 0} = \frac {1 0}{1 3} m / s ^ {2}$
$\frac {1}{2} \alpha t ^ {2} = \frac {1}{2} \times \frac {1 0 0}{1 3} \times \frac {1 0 0}{2 \pi} = \frac {5 0 0 0}{2 6 \pi} r \ell v$
$4 0 - T _ {A} = 4 a\tag{... (i}$
$T _ {B} - 2 0 = 2 a$
$(T _ {A} - T _ {B}) \times 0. 1 \frac {M (0 . 1) ^ {2}}{2} \alpha\tag{... (ii}$
... (iii)
$T _ {A} - T _ {B} = 2 \alpha\tag{... (vi}$
$a = \alpha R \Rightarrow a = \frac {\alpha}{1 0}\tag{... (iv}$
$\text { Now } (1) + (2) \Rightarrow T _ {B} - T _ {A} + 2 0 = 6 a$
$\therefore - 2 \alpha + 2 0 = 6 a$
$\therefore \quad - 2 \alpha + 2 0 = \frac {6 \alpha}{1 0} \Rightarrow 2 0 \frac {2 6 \alpha}{1 0}$
$\therefore \alpha = \frac {2 0 0}{2 6}$
$\therefore \alpha = \frac {1 0 0}{1 3}$
$\therefore \quad a = \frac {\alpha}{1 0} = \frac {1 0}{1 3} m / s ^ {2}$
$\frac {1}{2} \alpha t ^ {2} = \frac {1}{2} \times \frac {1 0 0}{1 3} \times \frac {1 0 0}{2 \pi} = \frac {5 0 0 0}{2 6 \pi} r \ell v$
JEE (Advanced) 2021
Q23
Allen
JEE Advanced PYQ
MSQ
One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height L above the hinge at point O. A block of weight $\alpha W$ is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2\sqrt{2})W$ . Which of the following statement(s) is(are) correct?
A.
The vertical component of reaction force at O does not depend on $\alpha$ .
B.
The horizontal component of reaction force at O is equal to W for $\alpha = 0.5$ .
C.
The tension in the rope is 2W for $\alpha = 0.5$ .
D.
The rope breaks if $\alpha > 1.5$ .
JEE (Main) 2020
Q24
Allen
JEE Main PYQ
MCQ
A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:
A.
$Mg\sqrt{1-\frac{a^{2}}{R^{2}}}$
B.
$Mg\sqrt{\left(\frac{R}{R-a}\right)^{2}-1}$
C.
$Mg\frac{a}{R}$
D.
$Mg\sqrt{1-\left(\frac{R-a}{R}\right)^{2}}$
JEE (Main) 2020
Q25
Allen
JEE Main PYQ
MCQ
A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. If the rod has mass M = 2 kg, and length l = 1 m, the value of $\theta$ is approximately :
(Take g = 10 m/s $^{4}$ )
A.
$6 9 ^ {\circ}$
B.
$6 3 ^ {\circ}$
C.
$5 5 ^ {\circ}$
D.
$4 9 ^ {\circ}$