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 as per the Class 11 Physics syllabus?
A rigid body is an idealised solid body where the distance between any two internal points remains constant, regardless of any external forces or torques applied to it. In simple terms, it's a body that does not deform or change shape during motion. While no real object is perfectly rigid, this concept is crucial for simplifying the analysis of motion in mechanics.
2. What is the primary difference between translational and rotational motion for a rigid body?
The primary difference lies in the paths of the constituent particles. In translational motion, all particles of the rigid body move along parallel paths, maintaining the same velocity. In rotational motion, all particles move in circular paths around a fixed axis, and each particle has the same angular velocity but different linear velocities depending on its distance from the axis.
3. What does the term 'rigid body dynamics' refer to?
Rigid body dynamics is the branch of mechanics that studies the motion of rigid bodies and the causes of that motion, which are forces and torques. It extends the principles of particle dynamics to solid objects by considering not only their translational movement but also their rotational movement, introducing key concepts like moment of inertia, torque, and angular momentum.
4. Why is the concept of a perfectly rigid body considered an idealisation?
The concept of a perfectly rigid body is an idealisation because, in reality, all materials deform to some extent when an external force is applied. Atoms and molecules shift their positions, causing a change in shape or size. However, for many practical applications in physics and engineering, this deformation is negligible, so we assume the body is rigid to make the mathematical analysis of its motion simpler and more manageable.
5. How are Euler's laws of motion important for rigid body dynamics?
Euler's laws are the rotational equivalent of Newton's laws of motion and are fundamental to rigid body dynamics. They describe the relationship between the forces, torques, and the resulting motion:
- The first law states that the rate of change of a body's linear momentum is equal to the net external force applied.
- The second law states that the rate of change of a body's angular momentum about a point is equal to the sum of the external torques about that same point.
These laws allow us to predict how a rigid body will move under any combination of forces and torques.
6. What is the role of the centre of mass in analysing rigid body motion?
The centre of mass is a crucial concept because it allows us to simplify the complex motion of a rigid body. The entire motion of the body can be described as a combination of two simpler motions: the translational motion of its centre of mass (as if all the mass were concentrated there) and the rotational motion of the body about its centre of mass. This separation makes problem-solving much more systematic.
7. Can you provide some real-world examples of objects treated as rigid bodies?
In many physics problems, common objects are treated as rigid bodies to simplify calculations, especially when their deformation is not relevant to the problem. Examples include:
- A spinning top or a gyroscope.
- Planets and moons orbiting in space.
- Wheels, gears, and shafts in machinery.
- A baseball bat or a cricket bat as it's being swung.
- Beams and girders used in building construction.
8. How does torque cause a rigid body to rotate?
Torque is the rotational equivalent of force. Just as a net force causes a body to accelerate linearly (F=ma), a net torque (τ) causes a rigid body to have an angular acceleration (α). The relationship is given by the equation τ = Iα, where 'I' is the moment of inertia. This means an applied torque will cause a stationary rigid body to start rotating or change the rotational speed of an already rotating body.
9. What distinguishes the dynamics of a rigid body from the dynamics of a point mass?
The key distinction is that a point mass can only undergo translational motion and is described by its position, velocity, and acceleration. A rigid body, being an extended object, can undergo both translational and rotational motion simultaneously. Therefore, its dynamics require additional concepts not applicable to a point mass, such as orientation, moment of inertia, torque, and angular momentum, to fully describe its state of motion.
