Rotational Motion

A wheel initially at rest undergoes uniform angular acceleration. In the first 2 s it rotates through θ₁ and in the next 2 s through θ₂. The ratio\(\frac{{\theta }_{2}}{{\theta }_{1}}\) is:

[JEE Main 2026, 5 Apr (Shift 2)]

If \(\vec{L}\) and \(\vec{P}\) represent the angular momentum and linear momentum respectively of a particle of mass ' \(m\) ' having position vector \(\vec{r}=\mathrm{a}(\hat{\mathrm{i}}\cos \omega \mathrm{t}+\hat{\mathrm{j}}\sin \omega \mathrm{t})\). The direction of force is:

Which of the following are correct expressions for torque acting on a body?
A. \(\vec{\tau }=\vec{\mathrm{r}}\times \vec{\mathrm{L}}\)
B. \(\vec{\tau }=\frac{\mathrm{d}}{\mathrm{dt}}(\vec{\mathrm{r}}\times \vec{\mathrm{p}})\)
C. \(\vec{\tau }=\vec{\mathrm{r}}\times \frac{\mathrm{d}\vec{\mathrm{p}}}{\mathrm{dt}}\)
D. \(\vec{\tau }=\mathrm{I}\vec{\alpha }\)
E. \(\vec{\tau }=\vec{\mathrm{r}}\times \vec{\mathrm{F}}\)
( \(\vec{\mathrm{r}}=\) position vector; \(\vec{\mathrm{p}}=\) linear momentum; \(\vec{\mathrm{L}}=\)angular momentum; \(\vec{\alpha }=\)angular acceleration; \(\mathrm{I}=\)moment of inertia; \(\vec{\mathrm{F}}=\) force; \(\mathrm{t}=\) time)
Choose the correct answer from the options given below :

[JEE Main 2025, 4 Apr (Shift 1)]

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

[JEE Main 2025, 24 Jan (Shift 2)]

Moment of inertia of a rod of mass ' M ' and length 'L' about an axis passing through its center and normal to its length is ' \(\alpha\) '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

[JEE Main 2025, 2 Apr (Shift 1)]

A rod of linear mass density ' \(\lambda\) ' and length ' L ' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is :

[JEE Main 2025, 8 Apr (Shift 1)]

The position of an object having mass 0.1 kg as a function of time t is given as \(\vec{r}=\left(10{t}^{2}\hat{i}+5{t}^{3}\hat{j}\right)\). At t=1 s, which of the following statements are correct?
A. The linear momentum \(\vec{p}=(2\hat{i}+1.5\hat{j})kgm/s\).
B. The force acting on the object \(\vec{F}=(2\hat{i}+3\hat{j})N\).
C. The angular momentum of the object about its origin \(\vec{L}=15\hat{k}Js\).
D. The torque acting on the object about its origin \(\vec{\tau }=20\hat{k}Nm\).

Choose the correct answer from the options given below:

[JEE Main 2026, 2 Apr (Shift 1)]

The torque due to the force \((2\hat{i}+\hat{j}+2\hat{k})\) about the origin, acting on a particle whose position vector is \((\hat{i}+\hat{j}+\hat{k})\), would be

[JEE Main 2025, 22 Jan (Shift 2)]

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is \(8\mathrm{m}/\mathrm{s}\). The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be :

A force \(\vec{\mathrm{F}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\text{ N}\) is acting at a point \((1,1,1)\). Find the torque of this force about the origin \((0,0,0)\).

(Shift II Memory Based)

A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination \(45^\circ .\) If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

[JEE Main 2025, 24 Jan (Shift 1)]

A solid sphere (A) of mass 5m and a spherical shell (B) of mass m, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of \({\mathrm{a}}_{\mathrm{A}}\) and \({\mathrm{a}}_{\mathrm{B}}\), respectively. The ratio of \({\mathrm{a}}_{\mathrm{A}}\) and \({\mathrm{a}}_{\mathrm{B}}\) is ____________.

[JEE Main 2026, 6 Apr (Shift 2)]

A solid cylinder of mass \(m\) and radius \(r\) is released from rest at the top of a rough inclined plane making an angle of \(45^{\circ}\) with the horizontal. Assuming the cylinder rolls without slipping, find the acceleration of the axis of the cylinder.

(Shift I - Memory Based)

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \({t}_{1}\) and \({t}_{2}\), respectively, then

[JEE Main 2025, 24 Jan (Shift 2)]

A disc of mass \(M\) and radius \(R\) is rotating about its axis. If the angle rotated about it as a function of time \(t\)t is \(\theta =a{t}^{2}+bt+c\), where \(a,b,\) and \(c\) are constants, find the power derived to the disc as a function of time.

A disc of mass \(M\) and radius \(R\) is rotating about its axis. If the angle rotated about it as a function of time \(t\)t is \(\theta =a{t}^{2}+bt+c\), where \(a,b,\) and \(c\) are constants, find the power derived to the disc as a function of time.

Two uniform discs of radius \(R\) and \(2R\) have moments of inertia \({I}_{1}\) and \({I}_{2}\)​ respectively about their central axes. If both discs have the same surface mass density, determine the ratio \({I}_{1}\mathrm{/}{I}_{2}\).

(Shift - I Memory Based)​

A solid sphere of mass ' \(m\) ' and radius ' \(r\) ' is allowed to roll without slipping from the highest point of an inclined plane of length ' \(L\) ' and makes an angle \(30^{\circ}\) with the horizontal. The speed of the particle at the bottom of the plane is \(v_1\). If the angle of inclination is increased to \(45^\circ\) while keeping \(L\) constant. Then the new speed of the sphere at the bottom of the plane is \(v_2\). The ratio \({v}_{1}^{2}:{v}_{2}^{2}\) is

[JEE Main 2025, 23 Jan (Shift 1)]

A solid sphere and hollow sphere rolls down purely equal distances on same inclined plane (starting from rest) in time \(t_1\) and \(t_2\) then

(Shift - II Memory based)

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

[JEE Main 2025, 2 Apr (Shift 2)]

A force \(\vec{\mathrm{F}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\text{ N}\) is acting at a point \((1,1,1)\). Find the torque of this force about the origin \((0,0,0)\).

(Shift II Memory Based)

A solid sphere and hollow sphere rolls down purely equal distances on same inclined plane (starting from rest) in time \(t_1\) and \(t_2\) then

(Shift - II Memory based)

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that \(\theta (t)=5{t}^{2}-8t\), where \(\theta (t)\) is the angular position of the rotating disc as a function of time \(t\).
How much power is delivered by the applied torque, at \(t=2\mathrm{s}\)?

[JEE Main 2025, 23 Jan (Shift 2)]

The torque due to the force \((2\hat{i}+\hat{j}+2\hat{k})\) about the origin, acting on a particle whose position vector is \((\hat{i}+\hat{j}+\hat{k})\), would be

A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination \(45^\circ .\) If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

[JEE Main 2025, 24 Jan (Shift 1)]

Two uniform discs of radius \(R\) and \(2R\) have moments of inertia \({I}_{1}\) and \({I}_{2}\)​ respectively about their central axes. If both discs have the same surface mass density, determine the ratio \({I}_{1}\mathrm{/}{I}_{2}\).

(Shift - I Memory Based)​

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \({t}_{1}\) and \({t}_{2}\), respectively, then

Torque on a uniform disk of mass \(2\text{ }\text{kg}\), radius \(1\text{ }\text{m}\), is given as \(\tau (t)=5{t}^{2}−8t\). If the disk was initially at rest, find the power by torque at \(t=1\text{ }\text{s}\).

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

[JEE Main 2025, 24 Jan (Shift 2)]

Two identical masses, each \(1\text{ }\text{kg}\), having velocity vectors:\(\vec{{\mathrm{V}}_{\mathrm{A}}}={\alpha }_{1}{\mathrm{t}}^{2}\hat{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\hat{\mathrm{j}}+{\alpha }_{3}\hat{\mathrm{k}},\) \(\vec{{\mathrm{V}}_{\mathrm{B}}}={\alpha }_{1}\hat{\mathrm{i}}+{\alpha }_{2}\hat{\mathrm{j}}+{\alpha }_{3}{\mathrm{t}}^{2}\hat{\mathrm{k}},\)where \({\alpha }_{1}=2,\text{ }{\alpha }_{2}=3n,\text{ }{\alpha }_{3}=4p\), and \(n,p\)are constants.

At \(t=1\text{ }\text{s}\), the velocities of \(A\) and \(B\) are orthogonal to each other and the magnitudes of their velocities are equal: \(\left|\vec{{\mathrm{V}}_{\mathrm{A}}}\right|=\left|\vec{{\mathrm{V}}_{\mathrm{B}}}\right|\).

Find the relative displacement between \(A\) and \(B\) at \(t=1\text{ }\text{s}\), and calculate the angular momentum of \(A\) with respect to \(B\).

A solid sphere of mass ' \(m\) ' and radius ' \(r\) ' is allowed to roll without slipping from the highest point of an inclined plane of length ' \(L\) ' and makes an angle \(30^{\circ}\) with the horizontal. The speed of the particle at the bottom of the plane is \(v_1\). If the angle of inclination is increased to \(45^\circ\) while keeping \(L\) constant. Then the new speed of the sphere at the bottom of the plane is \(v_2\). The ratio\({v}_{1}^{2}:{v}_{2}^{2}\) is

[JEE Main 2025, 23 Jan (Shift 1)]

A particle is rotating in a circular path and at any instant its motion can be described as \(\theta =\frac{5{t}^{4}}{40}-\frac{{t}^{3}}{3}\) The angular acceleration of the particle after 10 seconds is __________________ \(rad/{s}^{2}\).

[JEE Main 2026, 2 Apr (Shift 1)]

A solid sphere rolls without slipping on a horizontal plane. What is ratio of translational kinetic energy to the rotational kinetic energy of the sphere.

A solid sphere rolls without slipping on a horizontal plane. What is ratio of translational kinetic energy to the rotational kinetic energy of the sphere.

(Shift - II Memory based)

A solid sphere rolls without slipping on a horizontal plane. What is ratio of translational kinetic energy to the rotational kinetic energy of the sphere.

A body of mass \(m\) is projected with an initial velocity \({v}_{0}\) at an angle of \(4{5}^{∘}\) to the horizontal in the \(X−Y\) plane. Find the angular momentum of the body at the highest point with respect to the point of projection.

(Shift I Memory Based)​​

Two identical masses, each \(1\text{ }\text{kg}\), having velocity vectors:\(\vec{{\mathrm{V}}_{\mathrm{A}}}={\alpha }_{1}{\mathrm{t}}^{2}\hat{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\hat{\mathrm{j}}+{\alpha }_{3}\hat{\mathrm{k}},\) \(\vec{{\mathrm{V}}_{\mathrm{B}}}={\alpha }_{1}\hat{\mathrm{i}}+{\alpha }_{2}\hat{\mathrm{j}}+{\alpha }_{3}{\mathrm{t}}^{2}\hat{\mathrm{k}},\)where \({\alpha }_{1}=2,\text{ }{\alpha }_{2}=3n,\text{ }{\alpha }_{3}=4p\), and \(n,p\)are constants.

At \(t=1\text{ }\text{s}\), the velocities of \(A\) and \(B\) are orthogonal to each other and the magnitudes of their velocities are equal: \(\left|\vec{{\mathrm{V}}_{\mathrm{A}}}\right|=\left|\vec{{\mathrm{V}}_{\mathrm{B}}}\right|\).

Find the relative displacement between \(A\) and \(B\) at \(t=1\text{ }\text{s}\), and calculate the angular momentum of \(A\) with respect to \(B\).

Two simple pendulums having lengths \({l}_{1}\) and \({l}_{2}\) with negligible string mass undergo angular displacements \({\theta }_{1}\) and \({\theta }_{2}\), from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

[JEE Main 2025, 4 Apr (Shift 1)]

A body of mass \(m\) is projected with an initial velocity \({v}_{0}\) at an angle of \(4{5}^{∘}\) to the horizontal in the \(X−Y\) plane. Find the angular momentum of the body at the highest point with respect to the point of projection.

(Shift I Memory Based)​​

Torque on a uniform disk of mass \(2\text{ }\text{kg}\), radius \(1\text{ }\text{m}\), is given as \(\tau (t)=5{t}^{2}−8t\). If the disk was initially at rest, find the power by torque at \(t=1\text{ }\text{s}\).

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \({t}_{1}\) and \({t}_{2}\), respectively, then

[JEE Main 2025, 24 Jan (Shift 2)]

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are______ and _______ respectively.

[JEE Main 2026, 6 Apr (Shift 1)]

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that \(\theta (t)=5{t}^{2}-8t\), where \(\theta (t)\) is the angular position of the rotating disc as a function of time \(t\).
How much power is delivered by the applied torque, at \(t=2\mathrm{s}\)?

If \(\vec{L}\) and \(\vec{P}\) represent the angular momentum and linear momentum respectively of a particle of mass ' \(m\) ' having position vector \(\vec{r}=\mathrm{a}(\hat{\mathrm{i}}\cos \omega \mathrm{t}+\hat{\mathrm{j}}\sin \omega \mathrm{t})\). The direction of force is:

[JEE Main 2025, 4 Apr (Shift 1)]

A thin circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \(\omega\). If another disc of same dimensions but of mass \(M / 2\) is placed gently on the first disc co-axially, then the new angular velocity of the system is :

[JEE Main 2024, 08 Apr (Shift 2)]

A thin circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \(\omega\). If another disc of same dimensions but of mass \(M / 2\) is placed gently on the first disc co-axially, then the new angular velocity of the system is :

[JEE Main 2024, 08 Apr (Shift 2)]

A boy is rolling a \(0.5 kg\) ball on the frictionless floor with the speed of \(20 ms ^{-1}\). The ball gets deflected by an obstacle on the way. After deflection it moves with \(5 \%\) of its initial kinetic energy. What is the speed of the ball now? [JEE Main 2021, 17 Mar (Shift 1)]

Match List-I with List-II

Choose the correct answer from the options given below:

Match List-I with List-II

Choose the correct answer from the options given below:

[JEE Main 2021, 27 Jul (Shift 1)]

A force \( \vec{F}=(\hat{i}+2 \hat{j}+3 \hat{k}) N \) acts at a point \( (4 \hat{i}+3 \hat{j}-\hat{k}) m \). Then the magnitude of torque about the point \( (\hat{i}+2 \hat{j}+\hat{k}) \) \( m \) will be \( \sqrt{x} N-m \). The value of \( x \) is

[JEE Main 2020, 5 Sep (Shift 1)]

Two discs of moments of inertia \( I_{1} \) and \( I_{2} \) about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed \( \omega_{1} \) and \( \omega_{2} \) are brought into contact face to face with their axes of rotation coinciding with each other. What is the loss in kinetic energy of the system in the process?

[JEE Main 2017]

A bar of mass 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 ω. Which of the following statement is correct?