Example 1: In the figure shown below, the instantaneous speed of end A of the rod is 3m/s to the left. What will be the speed of end B of the rod at the given instant? Rigid Body Dynamics Class 11 Physics (A) 1m/s (B) 2m/s (C) 3m/s (D) 0m/s Answer: (C) 3m/s Explanation: Here, apply the concept of rigid bodies. We can see that A and B are on the same rod. The velocity of A and B along the line joining the two points is zero if they have no relative motion. vA cos30o = vB cos30o vA = vB vB = 3m/s (it is given that vA = 3m/s) Concept of Axis and Rotational Motion Rigid Body Dynamics Class 11 Physics ➤ Rotational Motion: Motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity. The dynamics for rotational motion are fully analogous to translational or linear dynamics. ➤ Pure rotational motion happens if all particles in the object move in a circular path about a single line (the axis of rotation). The radius vectors from the axis of rotation to every particle experience identical angular displacement simultaneously. ➤ Some of the examples of rotation about an axis are the rotation of the hand and the minute hand in the clocks, ceiling fan rotation, and closing and opening of doors. ➤ Every particle in the rotating body executes circular motion. ➤ Circular motion is a special case of rotational motion. ➤ Every point on an ideally spinning sphere is performing a circular motion about a centre that lies on a line. ➤ The centres of all of these circular motions will lie on a single line. ➤ Axis of rotation: A straight line is used as a reference to determine the position, symmetry and rotation. ➤ Axis of rotation is the locus of the centre of all circular motion. Concept of Axis ➤ Axis is a straight line about which a system or geometric body rotates. Example: the Earth’s axis. The body can be a small particle, smaller than a single molecule or atom. Or it could be a celestial body with a mass of a hundred and thousands of suns. Rotation around a fixed axis is a unique case of rotational motion. ➤ In the case of uniform cuboids, there are three types of standard axis of rotation. Rigid Body Dynamics Class 11 Physics Through the centre, along with the height. Rigid Body Dynamics Class 11 Physics Through the centre, along with the breadth. Rigid Body Dynamics Class 11 Physics Through the centre, along with the length. ➤ In the case of uniform discs, there are three types of the standard axis of rotation. Rigid Body Dynamics Class 11 Physics About the COM, the axis is perpendicular to the plane of the disc and passes through its centre. Rigid Body Dynamics Class 11 Physics About an axis along its diameter. Rigid Body Dynamics Class 11 Physics About an axis tangential to the circumference in the plane of the disc. ➤ In the case of thin uniform rods, there are two types of the standard axis of rotation. Rigid Body Dynamics Class 11 Physics About an axis passing through one of its ends and perpendicular to its length. Rigid Body Dynamics Class 11 Physics About an axis passing through its COM and perpendicular to its length. ➤ In the case of uniform cylinders, there are two types of the standard axis of rotation. Rigid Body Dynamics Class 11 Physics About its natural axis. Rigid Body Dynamics Class 11 Physics About an axis fixed at its rim along the length. Rigid Body Dynamics Class 11 Physics About an axis fixed at the centre of the base along the diameter. Different Types of Motion ➤ Pure Translational Motion It is a type of motion in which every point of a moving object moves uniformly in an identical direction. If a body is under translational motion, we can see that there is no variation in the orientation of the body. Translational motion is also known as translatory motion. Rigid Body Dynamics Class 11 Physics Here, in the same interval of time, every particle has Same displacement Same velocity Same acceleration ➤ Pure Rotational Motion Rotational motion can be stated as the motion of a body or system around a circular path in a constant orbit. The dynamics for rotational motion are entirely analogous to translational or linear dynamics. Most of the equations and formulas for the mechanics of rotating bodies are very similar to the motion equations and formulas for linear motion. Rigid Body Dynamics Class 11 Physics In this case, every particle moves In a circle whose centre lies on the axis of rotation. Through the same angle during a particular time interval (angular displacement, angular velocity, and angular acceleration are the same). ➤ Combination of Translational and Rotational Motion One significant principle of the combination of translational and rotational motion is that the kinetic energies of rotation and translation are additive. Particularly, we can calculate the net kinetic energy of an object by basically adding its translational and rotational kinetic energy. Rigid Body Dynamics Class 11 Physics In the case of a rolling ball, it possesses both translational and rotational motion. Moment of Inertia ➤ The moment of inertia is also called rotational inertia, second moment of mass, mass moment of inertia, or angular mass, of a rigid body is the quantity that describes the torque required for a particular angular acceleration associated with a rotational axis, related to how mass controls the forces required for a particular acceleration. It relies on the object’s mass distribution and the axis selected, with bigger moments needing more torque to vary the object’s rate of rotation. ➤ The moment of inertia can also be stated as the product of the section’s mass and the square of the length between the centroid and the reference axis of the section. ➤ All the particles in the rotating rigid body are performing circular motion with angular velocity . Rigid Body Dynamics Class 11 Physics ➤ The moment of inertia of a single point mass about an axis where the point mass is moving in a circular motion of radius r: Where r is the perpendicular distance of mass m from the axis of rotation. ➤ Moment of inertia of a system of n particles about an axis Moment of inertia is the rotational equivalent of mass in linear motion. ➤ Kinetic energy of all the particles in the rotating body is the sum of all kinetic energy of each particle in the body.