Rotational Motion

A uniform rod of length \(\ell\) is being rotated in a horizontal plane with a constant angular speed about an axis
passing through one of its ends. If the tension generated in the rod due to rotation is T(x) at a distance x from
the axis,...

An equilateral triangle ABC is cut from a thin solid sheet of wood. (see figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G ...

A slab is subjected to two forces \(\overrightarrow {{F_1}}\) and \(\overrightarrow {{F_2}}\) of same magnitude F as shown in the figure. Force \(\overrightarrow {{F_2}}\) is in XY-plane while force \(\overrightarrow {{F_1}}\) acts alo...

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular accele...

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5m, the angle between the force and the position vector is (in radians) :

A circular disc D1 of mass M and radius R has two identical discs D2 and D3 of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO' ,passing through the cent...

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is -

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coeffi...

A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its
instantaneous angular accel...

Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :

A particle of mass m is moving along a
trajectory given by
x = x0 + a cos\(\omega\)1t
y = y0 + b sin\(\omega\)2t
The torque, acting on the particle about the
origin, at t = 0 is :

A thin disc of mass M and radius R has mass
per unit area \(\sigma\)(r) = kr2 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 :

Two coaxial discs, having moments of inertia
I1 and I1/2, 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 wit...

A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of \({{7M} \over 8}\)
and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere.
Let I1 be ...

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

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the
figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25
rotations per second in 5s...

A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and z' axes will be :

A force of \(40\) \(N\) acts on a point \(B\) at the end of an \(L\)-shaped object, as shown in the figure. The angle \(\theta\) that will produce maximum moment of the force about point \(A\) is given by :

A uniform rod \(AB\) is suspended from a point \(X,\) at a variable distance \(x\) from \(A\), as shown, To make the rod horizontal, a mass \(m\) is suspended from its end \(A.\)A\(set of\)(m,x)$$ values is recorded. The appropriate varia...

A thin rod MN, free to rotate in the vertical plane aboutthe fixed end N, is held horizontal. When the end M is released the speed of this end, when the rod makes an angle \(\alpha\) with the horizontal, will be proportional to : (see figu...

A thin uniform bar of length \(L\) and mass \(8\) m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2\(\upsilon\) and \(\upsilon\) respective...

Seven identical circular planar disks, each of mass M and radius R are
welded symmetrically as shown. The moment of inertia of the arrangement
about the axis normal to the plane and passing through the point P is :

From a uniform circular disc of radius R and mass 9M, a small disc
of radius R/3 is removed as shown in the figure. The moment of
inertia of the remaining disc about an axis perpendicular to the plane
of the disc and passing through centre ...

A circular hole of radius \({R \over 4}\) is made in a thin uniform disc having mass M and radius R, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point O and perpendicular...

The machine as shown has 2 rods of length1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller g...

A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string as shown in the figure. The body is released from rest. Then the accele...

Moment of inertia of an equilateral triangular lamina ABC, about the axis
passing through its centre O and perpendicular to its plane is Io as shown in the figure. A cavity DEF is cut out from the lamina, where D, E, F are the mid points o...

In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?

The moment of inertia of a uniform cylinder of length \(l\) and radius R about its perpendicular bisector is \(I\).
What is the ratio \({l \over R}\) such that the moment of inertia is minimum?

A slender uniform rod of mass M and length \(l\) is pivoted at one end so that it
can rotate in a vertical plane (see figure). There is negligible friction at the
pivot. The free end is held vertically above the pivot and then released. The...

A cubical block of side 30 cm is moving with velocity 2 ms−1 on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular
velocity (in rad/s) of the block immediately after it hits the bump, is :

Concrete mixture is made by mixing cement, stone and sand in a rotating
cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum r...

A roller is made by joining together two cones at their vertices \(0\). It is kept on two rails \(AB\) and \(CD\), which are placed asymmetrically (see figure), with its axis perpendicular to \(CD\) and its center \(O\) at the center of lin...

A particle of mass m is moving along the
side of a square of side ‘a’, with a uniform
speed v in the x-y plane as shown in the
figure :

Which of the following statements is false
for the angular momentum \(\overrightarrow L\) about the
or...

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:

A mass \('m'\) is supported by a massless string wound around a uniform hollow cylinder of mass \(m\) and radius \(R.\) If the string does not slip on the cylinder, with what acceleration will the mass fall or release?

A bob of mass \(m\) attached to an inextensible string of length \(l\) is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega \,rad/s\) about the vertical. About the point of suspension:

A hoop of radius \(r\) and mass \(m\) rotating with an angular velocity \({\omega _0}\) is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop whe...

A pulley of radius \(2\) \(m\) is rotated about its axis by a force \(F = \left( {20t - 5{t^2}} \right)\) newton (where \(t\) is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation i...

A mass \(m\) hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass \(m\) and radius \(R.\) Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass \(m,\) if the st...

A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the jou...

A small particle of mass \(m\) is projected at an angle \(\theta\) with the \(x\)-axis with an initial velocity \({v_0}\) in the \(x\)-\(y\) plane as shown in the figure. At a time \(t < {{{v_0}\sin \theta } \over g},\) the angular moment...

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:

Consider a uniform square plate of side \(' a '\) and mass \('m'\). The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

Angular momentum of the particle rotating with a central force is constant due to

A round uniform body of radius \(R,\) mass \(M\) and moment of inertia \(I\) rolls down (without slipping) an inclined plane making an angle \(\theta\) with the horizontal. Then its acceleration is

For the given uniform square lamina \(ABCD\), whose center is \(O,\)

A force of \(- F\widehat k\) acts on \(O,\) the origin of the coordinate system. The torque about the point \((1, -1)\) is

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

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 rot...