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Rigid bodies are idealized depictions of bodies that don't change shape or deform. It is defined as a collection of a large number of particles, in which the distance between any two constituent particles remains fixed under the application of external force. Statics of Rigid bodies and dynamics of the rigid body were developed to answer a variety of problems that could not be explained using classical physics.
- Statics of Rigid bodies refers to the study of systems of interconnected bodies under no external forces i.e. objects that are either at rest or in constant motion.
- Dynamics of Rigid bodies refers to the study of the movement of systems of interconnected bodies under the action of external forces.
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Key Terms: Velocity, Acceleration, Equation of motions, Rigid bodies, Force, Motion, Mass, Translational motion, Rotational motion, Momentum, Angular velocity
What is a Rigid Body?
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A rigid body is one whose shape is precisely defined and unchangeable. It can be defined as a body where the distance between two given points on the body does not change when an external force is applied.
- A rigid body is usually considered as a continuous distribution of mass.
- Under the action of external force, there is zero or very small deformation on a rigid body.
- In simpler language, a rigid body is one that does not modify its shape when subjected to external forces.
- A perfectly rigid body does not exist in nature.
- According to the theory of special relativity, an object can only be supposed to be rigid if it is not moving near the speed of light.
- According to quantum mechanics, a rigid body is a collection of point masses. For example, molecules consisting of the point masses like electrons and nuclei are often seen as rigid bodies.
Read Also: Unit of Velocity
Rigid Body Dynamics
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Dynamics of Rigid bodies refers to the study of the movement of systems of interconnected bodies under the action of external forces. By assuming the bodies are rigid, the configuration of the system can be described by the translation and rotation of reference frames attached to the bodies.
A rigid body can undergo three forms of motion:
- Translational Motion
- Rotational Motion
- Combination of Translational and Rotational Motion
A rigid body's motion is either pure translation or a combination of translation and rotation if it is not pivoted or anchored in some way. Rotation is the motion of a rigid body that is pivoted or fixed in some way.
Translational Motion of a Rigid Body
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If any line drawn on the rigid body remains parallel to itself throughout the motion, then the body is said to be in pure translational motion. In pure translational motion, all of the body's particles have the same velocity at any one time.
We treat the entire system as a single point-like particle with mass m at the center of mass, travelling at the center of mass's velocity Vcm. At the center of mass, the external force acting on the system acts
\({\overrightarrow F_{ext}} = \frac{d\overrightarrow{P}_{sys}}{dt} = \frac{d}{dt}(m_T \overrightarrow{V}_{cm})\)
Where,
- Fext = External Force
- dP/dt = Change in momentum
- m = mass of the particle
- Vcm = velocity of the center of mass
Rotational Motion of a Rigid Body
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When a rigid body rotates around a fixed axis, each of its particles moves in a circle that exists in a plane perpendicular to the axis and has its center on the axis. If any line drawn on the rigid body does not remain parallel to itself throughout its motion, then the body is said to be in rotational motion.
To describe the rotational motion of the body, we will need certain parameters:
- The angle of rotation(θ): It is a measure of the amount of rotation of the body. When the body rotates, the angle by which any line drawn on the body rotates is the angle of rotation.
- Angular velocity(ω): The rate of rotation is measured by angular velocity. The angular velocity is defined as:
\(\omega = {d\theta\over dt}\)
- Angular acceleration(α): The angular acceleration is defined as the rate of change of angular velocity and it is given by
\(\alpha = {d\omega\over dt}\)
Equations of Rotational Motion for constant angular acceleration are given by:
- ω = ω0 + αt
- θ = ω0t + 1/2 αt2
- ω2 = ω02 + 2αθ
Read More: Angular Acceleration
Translational and Rotational Equations of Motion of a Rigid Body
Rolling motion is defined as a rigid body's combination of rotational and translational motion.
Kinetic Energy: We can get the total kinetic energy of a body by simply adding its rotational and translational kinetic energy. Thus the total kinetic energy of a particle is given by:
K = \(\frac{1}{2}\)Mvcm2 + \(\frac{1}{2}\)Iω2
Where,
- K= Kinetic Energy of the object
- M= mass of the object
- V= velocity of the object
- I = Moment of Inertia
- ω = Angular velocity
Things to Remember
- A rigid Body is a body whose shape doesn't change when an external force is applied to it.
- A rigid body can undergo three forms of motion: translational motion, rotational motion, and a combination of translational and rotational motion.
- In pure translational motion, all of the body's particles have the same velocity at any one time.
- When a rigid body rotates around a fixed axis, each of its particles moves in a circle that exists in a plane perpendicular to the axis and has its center on the axis is called the rotational motion of a rigid body.
- Rolling motion is defined as a rigid body's combination of rotational and translational motion.
Sample Questions
Ques. In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 A (1 Å = 10-10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus. (4 Marks)
Ans.
Let C.M. be at a distance x A from H-atom
Distance of C.M. from Cl atom = (1.27 -x) Å
Let the mass of H-atom = m units
The mass of the Cl-atom = 35.5 m units
If C.M. is taken at the origin, then
mx + (1.27 – x) 35.5 m
= 0 mx = – (1.27 – x) 35.5 m
A negative sign indicates that if Cl atom is on the right side of C.M. (+), the hydrogen atom is on
the left side of C.M. So, avoiding, if we get
x + 35.5x = 1.27 x 35.5
36.5x = 45.085
Therefore, the center of mass is located on the line joining H and Cl nuclei at a distance of 1.24 A from the H atom.
Ques. A rope of negligible mass is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping. (3 Marks)
Ans. Mass of hollow cylinder, M = 3 kg
Radius of hollow cylinder, R = 40 cm = 0.4 m
M.I. of the hollow cylinder about its axis
I = MR2 = 3 kg x (0.4 m)2 = 0.48 kg m2
Force F = 30 N
.’. Torque, τ=FxR = 30N x 0.4 m = 12 Nm
Ques. What are the essential features of angular momentum? (3 Marks)
Ans. The following are the key characteristics of angular momentum:
- The strength of a particle's rotational tendency around a point is determined by its angular momentum with respect to that point.
- The magnitude of angular momentum is determined by the particle's mass, velocity, and distance from the reference point, i.e. L = m v r.
- The rotational momentum vector notation is useful. Its axial direction is determined by the right-hand rule. L points in the direction perpendicular to the plane containing r and v.
Ques. What is the physical significance of M.I.? (2 Marks)
Ans. Rotational inertia, or M.I. of the body, is the incapacity of a body to change its condition of uniform rotation around an axis. It functions in rotatory motion in the same way as mass functions in translational motion, i.e. it is a rotating equivalent of mass.
Ques. A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with an initial angular speed equal to 10π rad s-1. Which of the two will start to roll earlier? The coefficient of kinetic friction is \(\mu k\) = 0.2. (5 Marks)
Ans. Given, Radii of the solid disc and the ring R = 10 cm = 0.1 m
Initial angular speed \(\omega_0 = 10 \pi\) rad s−1
Let, m be the mass of the disc/ring, and \(\omega\) be the final angular velocity after perfect rolling starts.
Also, the moment of inertia I = mk2,
where k is the radius of gyration.
As the net torque about the point ???? is zero, angular momentum is conserved about this point.
For ring,
Hence the disc begins to roll early than the ring.
Ques. Define Torque. What is its physical significance? (4 Marks)
Ans. Torque is defined as the turning effect that a force has on a rigid body.
i.e τ = Fd
where F = force applied on a body.
d is the ⊥ar distance of the line of action of the force from the axis of rotation.
Mathematically in vector form, τ may be expressed as
τ = r × F
i.e. it is the cross product of the position vector r and Force F
Ques. The speed of the inner layers of the whirlwind in a tornado is alarmingly high. Explain why? (2 Marks)
Ans. In a tornado, the inner layers of the whirlwind are close to the axis of rotation. It indicates that the M.I. of the air molecules in the inner layers is low.
As a result of the rule of conservation of angular momentum, the co of the inner layers of a tornado's whirlwind is extremely high.
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