In the subject of dynamics, rigid-body dynamics examines the motion of interconnected body systems under the influence of external forces. The assumption that the bodies are rigid and compact (they do not change shape under the influence of applied forces) streamlines analysis by skimming the parameters that describe the system’s configuration to the rotation and translation of reference frames connected to each body. This does not include objects that show fluid, plastic, and highly elastic behaviour. The idea of rigid body and rigid body dynamics arose to solve a wide range of mechanical problems that could not be analysed with old classical physics. Continuous movement, such as a fan’s rotation, a top, a potter’s wheel, etc., cannot be sufficiently analysed with a point mass. In practical scenarios, in the case of objects such as steel rods and wheels, bending or deformation is assumed to be negligible and treated as rigid.
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Important Rigid Body Class 11 Dynamics Concepts and Questions
Rigid Body
Rigid Body Definition
➤ A rigid body is a system of particles in which the distance between each pair of particles remains constant. Ideally, a rigid body is a body with an absolute, definite and unvarying shape. The lengths between every pair of inherent particles of such an object do not fluctuate.
➤ Take any two particles in a rigid body, and the distance between those particles will always remain the same.
➤ No relative motion between any pair of particles along the line joining them.
➤ An extended object in which deformation is neglected or zero.
➤ An extended object can be treated as a point mass centred at the COM in a circular/curvilinear motion.
➤ The shape of a rigid body cannot be changed. In other words, it cannot be compressed or stretched.
➤ The two forms of motion that rigid bodies undergo are translational motion and rotational motion.
➤ Moment of inertia is the capacity of the system to counter the variation generated in the rotational motion of a body.
➤ A macroscopic object constitutes a huge quantity of atoms. Explaining the movement of such a body with minimum simplification is impossible. Many bodies of interest, in fact, are clearly approximated by the supposition that the lengths between the atoms in the system are fixed.
➤ The centre of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero.
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?
(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
➤ 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.
Through the centre, along with the height.
Through the centre, along with the breadth.
Through the centre, along with the length.
➤ In the case of uniform discs, there are three types of the standard axis of rotation.
About the COM, the axis is perpendicular to the plane of the disc and passes through its centre.
About an axis along its diameter.
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.
About an axis passing through one of its ends and perpendicular to its length.
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.
About its natural axis.
About an axis fixed at its rim along the length.
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.
Here, in the same interval of time, every particle has
- Same displacement \(\begin{array}{l}\overrightarrow{S}\end{array} \)
- Same velocity \(\begin{array}{l}\overrightarrow{v}\end{array} \)
- Same acceleration \(\begin{array}{l}\overrightarrow{a}\end{array} \)
➤ 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.
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.
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
➤ 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:
\(\begin{array}{l}I = mr^2\end{array} \) |
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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
\(\begin{array}{l}I =\sum_{i=1}^{n} m_ir_i^2\end{array} \) |
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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.
\(\begin{array}{l}I = \sum_{i=1}^{n}m_iv_i^2\end{array} \) |
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Moment of inertia is the quantitative measure of rotational inertia of a body or angular mass. It depends on the total mass of the system and the mass distribution of the axis.
Moment of Inertia of Continuous Bodies
In reality, a rigid body will not be a system of discrete particles. Instead, it will be a system of continuous particles in continuation.
Let’s consider a body to have a continuous distribution of matter.
Moment of inertia of a small element,
Moment of inertia of the whole body,
\(\begin{array}{l}I = \int r^2dm \end{array} \) |
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➤ Uniform Rod
➤ Uniform Ring and Uniform Disc
➤ Rectangular Lamina and Equilateral Triangle
➤ Solid Cylinder and Hollow Cylinder
➤ Solid Sphere and Hollow Sphere
Example 2: A light rod of length l has two masses m1 and m2 attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is _________.
Explanation: Mass of the light rod is negligible and the mass approaches zero.
It is given that the axis is passing through the centre of mass. So, we can apply the condition of centre of mass.
That is,
Example 4: From a circular disc of radius R and mass 9M, a small disc of mass M and radius R/3 is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is _______.
Explanation:
Moment of inertia is an additive quantity. If we add a system of mass, then there will be a proportional increase in the moment of inertia. If we remove a system of mass, then there will be a proportional reduction in the moment of inertia.
Here, the net moment of inertia of the remaining disc is,
Parallel and Perpendicular Axis Theorem
➤ Perpendicular Axis Theorem
Moment of inertia is an additive quantity. If we add a system of mass, then there will be a proportional increase in the moment of inertia. If we remove a system of mass, then there will be a proportional reduction in the moment of inertia.
The perpendicular axis theorem states that a planar lamina’s moment of inertia (two-dimensional body) associated with an axis perpendicular to the lamina’s plane is equal to the sum of the moment of inertia associated with the two axes at right angles to one another, in its own plane meeting each other at the particular point where the perpendicular axis goes through it.
This theorem is applied when the object is symmetric in shape or structure about two out of the three axes. When the moment of inertia about the two axes is known, then the moment of inertia about the remaining third axis can be calculated using the standard equations.
The perpendicular axis theorem is not applicable for three-dimensional bodies like spheres, cylinders, cuboids, etc.
Let x and y be chosen in the plane of the body and z axis perpendicular to this plane, three axes being mutually perpendicular.
Perpendicular axis theorem states that,
\(\begin{array}{l}I_z + I_x = I_y\end{array} \) |
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This theorem is only applicable to two-dimensional bodies (lamina, ring, circular, etc.).
The point of intersection of the three axes need not be the centre of mass; it can be any point in the plane of the body which lies on the body or even outside it.
➤ Parallel Axis Theorem
Here, we have two axes which are parallel to each other. One of them is passing through the centre of mass, and the other one is not passing through the centre of mass.
The moment of inertia of a body about an axis parallel to an axis passing through its centre of mass and separated by perpendicular distance d is given by
\(\begin{array}{l}I = I_{com} + Md^2\end{array} \) |
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In engineering applications, we have to calculate a body’s moment of inertia. If the object is irregularly structured or shaped, then for calculating the moment of inertia in these scenarios, one can make use of the theorem of the parallel axis to find the moment of inertia at any section or point as long as the body’s centre of gravity is known. This is very useful in applied space physics, where the deriving of the moment of inertia of the satellites and spacecraft makes it viable for us to enter the outer planets and deep space.
Example 5: Consider a thin, uniform square sheet made of a rigid material. If its side is a, and mass m and moment of inertia I about one of its diagonals, then ________.
Explanation: Here, apply the perpendicular axis theorem since a thin uniform sheet is a type of laminar body.
Since this is a square, by symmetry, both diagonals are identical.
Example 6: Moment of inertia of a uniform circular disc of radius R and mass M about an axis passing from the edge of the disc and normal to the disc is ________.
Explanation: Since the axis is on the edge of the disk, apply the parallel axis theorem.
Radius of Gyration
➤ Any rigid body which is rotating about a given axis possesses some moment of inertia. The radius of gyration is basically the distance from the axis where the entire rigid body can be assumed to be concentrated, so that moment of inertia does not change.
➤ Distance from the axis that all mass can be concentrated to obtain the same mass ‘moment of inertia’ (I).
➤ The radius of gyration depends on
- Shape and size of the body.
- Position and configuration of the axis of rotation.
- Distribution of mass of the body with respect to the axis of rotation.
➤ The radius of gyration is independent of the mass of the body.
➤ Radius of Gyration of Common Bodies
Uniform Rod
Uniform Ring and Uniform Disc
Solid Cylinder and Hollow Cylinder
Solid Sphere and Hollow Sphere
Example 7: What will be the effect on the length of the day if the polar ice caps of the Earth melt?
(A) The moment of inertia decreases
(B) Radius of gyration decreases
(C) Radius of gyration increases
(D) Radius of gyration remains constant
Answer: (C) Radius of gyration increases
Explanation: We know that the Earth rotates on its own axis.
When the ice-caps in the polar region melt, the mass distribution changes. The total mass will be the same, but the mass distribution will change. This happens because the mass distribution depends on the shape. As ice caps melt, the shape of the ice structure drastically changes.
When ice-caps melt in the polar regions, the liquid water goes into the ocean. So, the ocean level will increase according to the rate of melting. Ocean levels in the hemispheres will be affected and increased. Melting of polar ice-caps will produce water spread around the Earth, going farther away from the axis of rotation. This will increase the radius of gyration.
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