Question

Show that the motion of a rigid body is determined at any instant by the velocities of three non-collinear points. Three particles \(A, B\), and \(C\) have velocities \(\boldsymbol{u}, \boldsymbol{v}\), and \(\boldsymbol{w}\) respectively relative to a frame \(R\). Show that they can belong to a rigid body if and only if $$ (a-b) \cdot(u-v)=(b-c) \cdot(v-w)=(c-a) \cdot(w-u)=0 $$ where \(\boldsymbol{a}, \boldsymbol{b}\), and \(\boldsymbol{c}\) are the position vectors of \(A, B\), and \(C\) from the origin of \(R\).

Short answer

Question: Explain the motion of a rigid body in terms of the velocities of three non-collinear points. Answer: The motion of a rigid body can be determined at any instant by the velocities of three non-collinear points if and only if the following condition is satisfied: $(a-b) \cdot(u-v)=(b-c) \cdot(v-w)=(c-a) \cdot(w-u)=0$. This condition ensures that the distance between any two points remains constant, which is a requirement for the points to belong to a rigid body.

Step-by-step solution

Condition for particles to belong to a rigid body

For three particles A, B, and C to belong to a rigid body, the distance between any two points should be constant. To find the distance between points A and B, we can use the position vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\). The vector resulting from subtracting \(\boldsymbol{b}\) from \(\boldsymbol{a}\), \(\boldsymbol{a} - \boldsymbol{b}\), will have the magnitude equal to the distance between A and B, and similarly for the other pairs. Since the distance is constant, the rate of change of this distance must be zero.

Evaluating the rate of change of distance

To evaluate the rate of change of distance between the points, we can differentiate the magnitude of the vectors mentioned in step 1 with respect to time. Let's consider the first pair of points, A and B. The rate of change of the vector \(\boldsymbol{a} - \boldsymbol{b}\) is $$ \frac{\mathrm{d}}{\mathrm{dt}}(\boldsymbol{a}-\boldsymbol{b}) = \frac{\mathrm{d}\boldsymbol{a}}{\mathrm{dt}} - \frac{\mathrm{d}\boldsymbol{b}}{\mathrm{dt}} = \boldsymbol{u} - \boldsymbol{v} $$ Now, we can evaluate the dot product with the position vector as follows: $$ (\boldsymbol{a}-\boldsymbol{b}) \cdot (\boldsymbol{u}-\boldsymbol{v})=0 $$ We can perform the same steps for the other pairs of points. This will result in three conditions: $$ (\boldsymbol{a}-\boldsymbol{b}) \cdot (\boldsymbol{u}-\boldsymbol{v})=0 \\ (\boldsymbol{b}-\boldsymbol{c}) \cdot (\boldsymbol{v}-\boldsymbol{w})=0 \\ (\boldsymbol{c}-\boldsymbol{a}) \cdot (\boldsymbol{w}-\boldsymbol{u})=0 $$

Comparing the obtained conditions to the given condition

We have derived the conditions for the particles A, B, and C to belong to a rigid body: $$ (\boldsymbol{a}-\boldsymbol{b}) \cdot (\boldsymbol{u}-\boldsymbol{v})=0 \\ (\boldsymbol{b}-\boldsymbol{c}) \cdot (\boldsymbol{v}-\boldsymbol{w})=0 \\ (\boldsymbol{c}-\boldsymbol{a}) \cdot (\boldsymbol{w}-\boldsymbol{u})=0 $$ These conditions are the same as the given condition: $$ (a-b) \cdot(u-v)=(b-c) \cdot(v-w)=(c-a) \cdot(w-u)=0 $$ Since these conditions are satisfied, it implies that the particles A, B, and C can belong to a rigid body. Thus, we have shown that the motion of a rigid body is determined at any instant by the velocities of three non-collinear points if and only if the given condition is satisfied.

Key concepts

Analytical Dynamics

In the realm of physics, specifically under the domain of classical mechanics, analytical dynamics plays a crucial role in understanding how objects move. It deals with the study of particles and rigid bodies in motion, and the forces acting upon them.

Analytical dynamics looks at the relationships that govern motion using differential equations, representing physical quantities like velocity and acceleration. One of the primary equations used in this field is Newton's second law, which states that the force on an object is equal to its mass times its acceleration (\( F = ma \)). When we apply this law to a rigid body, it allows us to understand how the body moves as a whole, even though it is made up of many particles.

Velocity of Particles

The velocity of a particle is a vector quantity that represents the rate of change of its position with respect to time. It not only describes how fast the particle is moving but also in what direction.

When we are looking at multiple particles in a system, such as points A, B, and C in a rigid body, their individual velocities will combine to give us a clear picture of the body's overall motion. This is significant in analytical dynamics, as the coordinated velocity of each particle is necessary to ensure that the body moves without changing its shape, a key characteristic of rigid body motion.

Constant Distance

A fundamental concept in the study of rigid bodies is that the distance between any two particles within the body remains constant over time. This notion is inherent in the definition of a rigid body; it does not deform or change shape.

In terms of mathematical representation, the constancy of distance is translated into the requirement that the magnitude of the vector difference between position vectors of any two particles should not change. This must be true for any pair of particles in the body, ensuring the body's shape stays unchanged during motion.

Rate of Change of Distance

In a dynamic scenario, we are interested in how quantities change over time. The rate of change of distance between particles in a rigid body gives us vital information about the body's movement. For a rigid body, as previously mentioned, this rate must be zero since the distances are constant.

Analytically, we can gauge this rate by differentiating the vector difference of position vectors with respect to time. If this derivative equals zero, it confirms that the particles maintain a constant distance apart during motion. This principle is fundamental to proving the condition for the rigid body behavior of a system of particles. By ensuring the dot product of the differentiated position vectors and the velocities equals zero, we maintain the condition for constant distance, which validates the behavior of a body as a rigid entity.