Rigid Body and Rigid Body 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:

1. 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

2. 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.

3. 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

1. 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)

2. 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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3. 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

4. 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