Rigid Dynamics
The property of a rigid body can be understood through an example discussed below:
Consider a body, assume two internal points separated by a distance d.
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From Fig.1
If this distance d between point A₀ and B₀ does not change, then this body is rigid.
Practically, a perfectly rigid body doesn’t exist.
However, in rotational motion, bodies like a sphere, rods are considered rigid bodies. i.e. each body will have two internal points with a fixed distance in itself.
Now, dynamics is that region because of which motion occurs. Here, when we talk of a force, dynamics come into play. Therefore, the dynamics of a rigid body are called rigid body dynamics.
Rigid Body Dynamics
A body, in general, can execute both translational and rotational motions.
For a body in translational motion:
Consider a body with two internal points separated by some distance.
Now, when we join these two points in a rigid body, as shown in Fig.2 below:
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From Fig.2 (a)
The line AB joins these two internal points A and B. Now, the line AB and A’B’ remain parallel during the motion. We can say that a body is said to be in translational motion when the line joining the two internal points before and during the motion remains parallel.
Here, all the particles in the line AB continue to move in parallel, before and during the motion.
Hence, the path is straight, so that’s why it's a rectilinear translational motion.
From Fig.2(b)
Here, we can see that the lines are still parallel to itself before and during the motion. Therefore, the path of particles is also parallel.
However, the curvilinear motion is happening. So, it’s called the curvilinear translational motion.
Now, if we consider a body rotating about its axis. Look at the Fig.3 below:
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All the internal particles move in a circular path about a fixed point or axis. When we join point D₁ to D₂ and point P₁ to P₂. Then lines D₁D₂ and P₁P₂ aren’t parallel to each other. Such a kind of motion of a rigid body is called rotational motion.
Dynamics Rigid Body Kinematics
Applying Newton’s laws of motion in rotational motion:
A body continues to be in a state of rest or a uniform rotational motion about a fixed axis unless an external torque is applied to it.
According to Newton’s second law of motion:
When a force is applied to a body of mass m, it starts accelerating in the direction of motion.
So, the equation is given by,
∑F = m ∑a
Now, if we calculate acceleration concerning a frame (other than inertial frame because points in this frame have no acceleration) that is purely translating with an acceleration a₀. Then the equation will be written as:
∑F = m ∑a + m ∑a₀
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Rigid Body Equations of Motion
The laws of motion for a rigid body are called Euler’s laws.
The two laws are relative to the inertial frame of reference, stated as:
For the translational motion:
∑ F = d/dt(G)
∑ M₀ = d/dt(H₀)
Here, O is the fixed point on the inertial frame of reference. G is the linear momentum of a particle given by,
G = mv
For the rotational motion:
∑ F = d/dt(G)...(3)
∑ Mₒₘ = d/dt(Hₒₘ)..(4)
Here, fixed point O on an inertial frame is obtained by the center of mass ₒₘ.
Consider an arbitrary point P in place of the center of mass (ₒₘ).
∑ F = d/dt(G)
∑ Mₚ = d/dt(Hₒₘ) + r ₚ/ₒₘ x d/dt(G)
Here, r ₚ/ₒₘ is the position of the center of mass relative to the selected point P.
Rigid Body Dynamics Equations
Linear momentum of a body
For a body having particles, linear momentum will be the sum of the G of its particles.
If a body has particles each having mass Δmᵢ moving with velocity Δvᵢ. Then,
G = ∑ Δ mᵢvᵢ
If vₒₘ is the velocity of the center of mass. Then by Euler’s law:
As we know ∑F = maₒₘ
∑F = d/dt(mvₒₘ)
Angular momentum of a body for its particles
If Hₒ denotes the angular momentum of a body, then:
Hₒ = ∑rᵢ x Δ mᵢvᵢ
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The relation of the moment of forces when they are taken at different points
The moment of a force considered at O can be related to the moment of the
same force taken about point A is represented as:
Mₒ = Mₐ + rₐ/ₒ x F
The relation between angular momentum when taken at different points
Hₒ = ∫ r x vdm = ∫ (ρ + rₐ/ₒ) x vdm
m m
= ∫ ρ x vdm + ∫ rₐ/ₒ x vdm
m m
= Hₐ + rₐ/ₒ x mₒₘ
FAQs on Rigid Body and Rigid Body Dynamics
1. What is a rigid body in the context of Class 11 Physics?
In Class 11 Physics, a rigid body is defined as an idealised solid object in which the distance between any two internal points remains constant, irrespective of the external forces applied. This means the body does not deform, bend, or change its shape during motion. While no real object is perfectly rigid, this concept is fundamental for simplifying the study of mechanics.
2. What is the key difference between translational and rotational motion for a rigid body?
The key difference lies in the motion of the body's constituent particles. In pure translational motion, all particles of the body move with the same velocity along parallel paths. In pure rotational motion, all particles move in circles around a fixed axis, sharing the same angular velocity but having different linear velocities depending on their distance from the axis.
3. How does 'rigid body dynamics' differ from the dynamics of a simple point mass?
The dynamics of a point mass only considers translational motion, described by its mass, position, and velocity. Rigid body dynamics is more complex because an extended body can undergo both translational and rotational motion simultaneously. Therefore, it requires additional concepts to describe its state, such as the centre of mass, moment of inertia, torque, and angular momentum.
4. What are the main topics studied under rigid body dynamics?
Rigid body dynamics is the branch of mechanics that studies the causes of motion in rigid bodies. Key topics include:
- The concept of the centre of mass.
- Moment of inertia, which measures rotational inertia.
- Torque, the rotational equivalent of force.
- Angular momentum and its conservation.
- Theorems of parallel and perpendicular axes for calculating the moment of inertia.
- Analysis of complex motions like rolling motion (a combination of translation and rotation).
5. Why is the concept of a 'perfectly rigid body' an important idealisation in physics?
The concept is an important idealisation because real objects always deform slightly under external forces. However, in many practical scenarios, this deformation is negligible and doesn't significantly affect the overall motion. By assuming the body is perfectly rigid, we can ignore these complex internal forces and deformations, which makes the mathematical analysis of its motion, especially rotation, much simpler and more manageable.
6. What is the significance of the centre of mass when analysing the motion of a rigid body?
The centre of mass is a unique point that simplifies the analysis of complex rigid body motion. The entire motion can be separated into two parts: the translational motion of the centre of mass (as if all the body's mass and external forces were concentrated at this point) and the rotational motion of the body about its centre of mass. This separation makes solving problems much more systematic.
7. How does torque create rotational motion in a rigid body?
Torque (τ) is the rotational analogue of force. Just as a net force causes linear acceleration (F = ma), a net torque causes an object to have an angular acceleration (α). This relationship is described by the equation τ = Iα, where 'I' is the moment of inertia. An applied torque will cause a stationary body to start rotating or change the angular velocity of an already rotating body.
8. Are 'rotational motion' and 'rigid body dynamics' the same thing?
No, they are related but distinct concepts. Rotational motion is a specific type of movement where an object spins around an axis. Rigid body dynamics is the comprehensive field of study that analyses all motions of a rigid body, including pure rotation, pure translation, and complex combinations like rolling. It uses principles like torque and moment of inertia to explain the causes behind these motions.
9. What is the moment of inertia and why is it also called rotational inertia?
The moment of inertia (I) is a physical quantity that measures a rigid body's resistance to a change in its state of rotational motion. It is called rotational inertia because it plays the same role in rotation that mass (linear inertia) plays in translation. A larger moment of inertia means it is harder to start or stop the object's rotation, just as a larger mass makes it harder to start or stop its linear motion.
10. What are some real-world examples where objects are treated as rigid bodies?
In many physics and engineering problems, objects are treated as rigid bodies to simplify analysis. Common examples include:
- Planets, moons, and asteroids in orbital mechanics.
- A spinning top, a gyroscope, or a bicycle wheel.
- Gears, shafts, and flywheels in machinery.
- A cricket bat or a golf club during a swing.
- Structural components like beams and girders in construction analysis.
11. If a rigid body is in pure rotational motion, do all its particles have the same linear velocity?
No. While all particles of a rigid body in pure rotation have the same angular velocity (ω), their linear velocity (v) is different. The linear velocity of any particle is given by the formula v = rω, where 'r' is the perpendicular distance of the particle from the axis of rotation. Therefore, particles farther from the axis move faster (have a higher linear velocity) than particles closer to the axis.
