Question

Establish the properties of the line \(L\) defined by \((1.41)\) for the general motion of a rigid body. That is, show that if \(P\) is a point of \(L\), then \(\boldsymbol{v}_{P}\) is proportional to \(\omega\); and show conversely that if \(v_{P}\) is proportional to \(\omega\), then \(P\) lies on \(L\). (Assume that \(\omega \neq 0\).)

Short answer

Question: Prove that for the general motion of a rigid body, a point P lies on the line L if and only if its velocity vector \(\boldsymbol{v}_{P}\) is proportional to the angular velocity vector \(\boldsymbol{\omega}\). Answer: If P lies on the line L, then \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\). Conversely, if \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\), then point P must lie on the line L.

Step-by-step solution

Determine the Line L and Velocity Vector \(v_P\)

The line L is defined by (1.41) and represents the axis of rotation. For an arbitrary point \(P\) on the line \(L\), the velocity vector can be described as \(\boldsymbol{v}_{P} = \boldsymbol{\omega} \times \boldsymbol{r}_{P}\), where \(\boldsymbol{r}_{P}\) is the position vector of the point \(P\) and \(\boldsymbol{\omega}\) is the angular velocity vector.

Show that \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\) for a point on L

If point \(P\) is on line \(L\), then the line \(L\) is parallel to the position vector \(\boldsymbol{r}_{P}\). Since they are parallel, the cross product of \(\boldsymbol{\omega}\) and \(\boldsymbol{r}_P\) will also be parallel to \(\boldsymbol{\omega}\), which means \(\boldsymbol{v}_{P} = k\boldsymbol{\omega}\) for some constant \(k\). This shows that if \(P\) is a point of \(L\), then \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\).

Show that if \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\), then \(P\) lies on L

Now we need to show the converse. If \(\boldsymbol{v}_P\) is proportional to \(\boldsymbol{\omega}\), then there is a constant \(k\) such that \(\boldsymbol{v}_P = k\boldsymbol{\omega}\). Rearranging this equation, we obtain \(\boldsymbol{v}_{P} - k\boldsymbol{\omega} = 0\). Taking the cross product of this equation with the angular velocity vector \(\boldsymbol{\omega}\), we get \((\boldsymbol{v}_{P} - k\boldsymbol{\omega}) \times \boldsymbol{\omega} = 0\). Expanding the cross product, we have \(\boldsymbol{v}_{P} \times \boldsymbol{\omega} - k\boldsymbol{\omega} \times \boldsymbol{\omega} = 0\). Since the cross product of a vector with itself is always zero, the equation simplifies to \(\boldsymbol{v}_{P} \times \boldsymbol{\omega} = 0\). Recall the velocity vector formula, \(\boldsymbol{v}_{P} = \boldsymbol{\omega} \times \boldsymbol{r}_{P}\). Substituting this into the equation, we get \((\boldsymbol{\omega} \times \boldsymbol{r}_{P}) \times \boldsymbol{\omega} = 0\). Using the vector triple product identity, \(\boldsymbol{A} \times (\boldsymbol{B} \times \boldsymbol{C}) = (\boldsymbol{A} \cdot \boldsymbol{C})\boldsymbol{B} - (\boldsymbol{A} \cdot \boldsymbol{B})\boldsymbol{C}\), we can rewrite the above equation as \((\boldsymbol{\omega} \cdot \boldsymbol{\omega})\boldsymbol{r}_{P} - (\boldsymbol{\omega} \cdot \boldsymbol{r}_{P})\boldsymbol{\omega} = 0\). Or equivalently, \(\boldsymbol{r}_{P} = \frac{\boldsymbol{\omega} \cdot \boldsymbol{r}_{P}}{\boldsymbol{\omega} \cdot \boldsymbol{\omega}}\boldsymbol{\omega}\). This equation implies that the position vector \(\boldsymbol{r}_{P}\) is parallel to the angular velocity vector \(\boldsymbol{\omega}\). Therefore, point \(P\) must lie on the line of action \(L\). Thus, we have shown that if \(\boldsymbol{v}_{P}\) is proportional to \(\boldsymbol{\omega}\), then \(P\) lies on \(L\).

Key concepts

Rigid Body Motion

Rigid Body Motion refers to the movement of a solid body that does not deform during its motion. This means the distances between any two points on the body remain constant throughout the motion, implying no internal strain, only displacement.

• Rigid bodies can have two types of motion: translational (where all parts of the body move in the same direction and by the same amount) and rotational (where the body rotates around a fixed axis).
• In problems involving rigid body motion, it is often important to determine the relationship between various physical quantities such as force, torque, and angular momentum.

Understanding rigid body motion is crucial because it allows us to analyze complex systems, such as engines or machinery, where the motion can be decomposed into simpler rigid body motions. This simplification is possible because each part of the machinery behaves as a rigid body.

Axis of Rotation

The Axis of Rotation is an imaginary line around which a rigid body rotates. In geometrical terms, this line is called a straight line, but in practical scenarios, it can be visually represented as a rod around which an object spins. The key characteristics of this axis include:

• For any rigid body undergoing rotational motion, every point moves in a circular path around the axis of rotation.
• The axis of rotation can either be inside the body or external to it, like a planet rotating around the Sun.

When analyzing problems involving rotation, we often consider only the motion relative to this axis to simplify calculations. By focusing on the axis, we can determine other important attributes such as angular displacement and angular momentum.

For instance, in the exercise at hand, the axis of rotation is essential for defining the line \(L\), where any point \(P\) on this line would mean that the velocities and movements are directly correlated with the angular velocity.

Angular Velocity

Angular Velocity is a vector quantity that measures how fast an object rotates or revolves relative to another point, typically the center of rotation. This quantity is measured in radians per second (rad/s).

• It not only tells us the speed of rotation but also the direction in which the rotation is occurring.
• For rigid body motion, angular velocity remains constant if there are no external torques acting on the body.

Angular velocity plays an integral role when analyzing rotational movements, such as those addressed in the given problem. It helps in quantifying the relationship between any point's velocity on the body and its distance from the center of rotation.

The direction of the angular velocity vector, \(\boldsymbol{\omega}\), is determined by the right-hand rule: if the fingers of the right hand are curled in the direction of rotation, the thumb points in the direction of the angular velocity vector.

Vector Analysis

Vector Analysis is a branch of mathematics concerned with quantities that have both magnitude and direction. Vectors are essential in physics and engineering for describing forces, velocities, and other vector quantities.

• Vectors are typically represented graphically as arrows, the length representing magnitude, and the direction of the arrow indicating the vector's direction.
• Operations such as addition, subtraction, and cross product are used to analyze and modify vectors.

In the context of rigid body dynamics, vector analysis is crucial. For instance, the cross product \(\boldsymbol{\omega} \times \boldsymbol{r}_{P}\) in the exercise establishes the relationship between the position vector and the velocity of points on a rigid body. This product helps determine how the body's rotation affects different points in space.

Understanding how to manipulate and interpret vectors allows us to solve complex dynamics problems, as they often involve determining unknown quantities based on known vectors and their interactions.