Rotational Motion

A solid sphere of mass m and radius R is placed on a smooth horizontal surface. A sudden blow is given horizontally to the sphere at a height h = 4R/5 above the centre line. If I is the impulse of the blow then find (A) the minimum time after which the highest point B will touch the ground (B) the displacement of the centre of mass during this internal.

Two uniform cylinders, each of mass m = 10 kg and radius r = 150 mm, are connected by a rough belt as shown. If the system is released from rest, determine: (A) the velocity of the centre of cylinder A after it has moved through 1.2 m. (B) the tension in the portion of the belt connecting the two cylinders.

The free end of the string wound on the surface of a solid cylinder of mass M and radius r is pulled up by a force F as shown. If there is sufficient friction between cylinder and floor so that the cylinder is able to roll without slipping, find the maximum angular acceleration that the cylinder can have (without cylinder leaving contact with surface).

A uniform disk of mass m and radius R is projected horizontally with velocity $v_{0}$ on a rough horizontal floor so that it starts off with a purely sliding motion at t = 0. After $t_{0}$ seconds it acquires a purely rolling motion as shown in figure. (I) Calculate the velocity of the centre of mass of the disk at $t_{0}$ . (II) Assuming the coefficient of friction to be $\mu$ calculate $t_{0}$ . Also calculate the work done by the frictional force as a function of time and the total work done by it over a time t much longer than $t_{0}$ .

Water flowing along an open channel drives an undershot waterwheel of radius 2 m (figure). The water approaches the wheel with a speed of 5.0 m/s and leaves with a speed of 2.5 m/s; the amount of water passing by is 200 kg per second. At what rate does the water deliver angular momentum to the wheel (in kJ)?

A solid uniform disk of mass m rolls down a fixed inclined plane without slipping with an acceleration a. Find the frictional force on the disk due to surface of the plane:

Portion AB of the wedge shown in figure is rough and BC is smooth. A solid cylinder rolls without slipping from A to B. Find the ratio of translational kinetic energy to rotational kinetic energy, when the cylinder reaches point C.

A solid cylinder C and a hollow pipe P of same diameter are in contact when they are released from rest as shown in the figure on a long incline plane. Cylinder C and pipe P roll without slipping. Determine the clear gap (in m) between them after 6 seconds.

A uniform rod of length l is given an impulse at right angle to its length as shown. Find the distance of instantaneous centre of rotation from the centre of the rod.

A uniform rod of mass m and length $\ell$ is placed in gravity free space and linear impulse J is given to the rod at a distance $x = \ell/4$ from centre 'C' and perpendicular to the rod. Point A is at a distance $\ell/3$ from centre as shown in the figure. Then find
(I) Speed of centre of rod
(II) Speed of point A
(III) Speed of upper end of rod
(IV) Speed of lower end of rod

A uniform rod AB of length L and mass m is suspended freely at A and hangs vertically at rest when a particle of same mass m is fired horizontally with speed v to strike the rod at its mid point. If the particle comes to rest after the impact, then find the impulsive reaction at A.

On a smooth table two particles of mass m each, travelling with a velocity $v_{0}$ in opposite directions, strike the ends of a rigid massless rod of length l, kept perpendicular to their velocity. The particles stick to the rod after the collision. Find the tension in rod during subsequent motion.

A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end 'A' of the rod with a velocity $v_{0}$ in the direction perpendicular to AB. The collision is completely elastic. After the collision the particle comes to rest. (A) Find the ratio $m / M$ . (B) A point P on the rod is at rest immediately after the collision. Find the distance AP.

A thin uniform straight rod of mass 2 kg and length 1 m is free to rotate about its upper end when at rest. It receives an impulsive blow of 10 Ns at its lowest point, normal to its length as shown in figure. Find the kinetic energy of rod just after impact.

A uniform rod of length 0.72 m is pivoted at one end and is hanging as shown. It receives a horizontal impulse at a distance x below the pivot as shown. If impulse applied by the pivot is zero, then find the value of x (in cm).

A mass m travelling at speed $v_{0}$ strikes perpendicularly to a stick of mass m and length $\ell = 1.2m$ , which is initially at rest as shown in the figure. The mass collides completely inelastically with the stick at one of its ends, and sticks to it. The location of point from end A (in cm) which is at instantaneous rest just after collision is $x_{0}$ . Value of $x_{0}/10$ is

Figure shows a vertical force applied tangentially to a uniform cylinder of weight $F_{g}$ . The coefficient of static friction between the cylinder and both surfaces is 0.500. In terms of $F_{g}$ , find the maximum force P that can be applied that does not cause the cylinder to rotate.

Three particles A, B, and C each of mass m, are connected to each other by three massless rigid rods to form a rigid, equilateral triangular body of side $\ell$ . This body is placed on a horizontal frictionless table (x-y plane) and is hinged to it at the point A, so that it can move without friction about the vertical axis through A (see figure). The body is set into rotational motion on the table about A with a constant angular velocity $\omega$ . (A) Find the magnitude of the horizontal force exerted by the hinge on the body. (B) At time T, when the side BC is parallel to the x-axis, a force F is applied on B along BC (as shown). Obtain the x-component and the y-component of the force exerted by the hinge on the body, immediately after time T.

A man pushes a cylinder of mass $m_{1}$ with help of a plank of mass $m_{2}$ as shown. There is no slipping at any contact. the horizontal component of the force applied by the man is F. Find: (A) the accelerations of the plank and the centre of mass of the cylinder, and (B) the magnitudes and directions of frictional forces at contact points.

Two thin circular disks of mass 2 kg and radius 10 cm each are joined by a rigid massless rod of length 20 cm. The axis of the rod is along the perpendicular to the planes of the disks through their centre. The object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of motion of the truck. Its friction with the floor of the truck is large enough so that the object can roll on the truck without slipping. Take x-axis as the direction of motion of the truck and z-axis as the vertically upwards direction. If the truck has an acceleration of $9 \, m/s^{2}$ calculate : (A) the force of friction on each disk. (B) the magnitude and the direction of the frictional torque acting on each disk about the centre of mass O of the object. Express the torque in the vector form of unit vectors in the x-y and z directions.

Two heavy metallic plates are joined together at $90^{\circ}$ to each other. A laminar sheet of mass 30 Kg is hinged at the line AB joining the two heavy metallic plates. The hinges are frictionless. The moment of inertia of the laminar sheet about an axis parallel to AB and passing through its centre of mass is $1.2\ Kg\cdot m^{2}$ . Two rubber obstacles P and Q are fixed, one on each metallic plate at a distance 0.5 m from the line AB. This distance is chosen so that the reaction due to the hinges on the laminar sheet is zero during the impact. Initially the laminar sheet hits one of the obstacles with an angular velocity 1 rad/s and turns back. If the impulse on the sheet due to each obstacle is 6 N-s. (A) Find the location of the centre of mass of the laminar sheet from AB. (B) At what angular velocity does the laminar sheet come back after the first impact? (C) After how many impacts, does the laminar sheet come to rest?

A particle starts from the point $(0m, 8m)$ and moves with uniform velocity of $3\hat{i}m/s$ . After 5 seconds, the angular velocity of the particle about the origin will be:

Two points of a rigid body are moving as shown. The angular velocity of the body is:

A non-uniform sphere can be kept on a rough inclined plane so that it is in equilibrium. In the figure below the dots represents location of centre of mass. In which one of the positions can sphere be in equilibrium?

A uniform flag pole of length L and mass M is pivoted on the ground with a frictionless hinge. The flag pole makes an angle $\theta$ with the horizontal. The moment of inertia of the flag pole about one end is $(1/3)ML^{2}$ . If it starts falling from the position shown in the accompanying figure, the linear acceleration of the free end of the flag pole (labeled P) immediately after it starts falling off would be: