To tackle the problem of finding the value of the dot product \( \mathbf{A} \cdot \mathbf{B} \), where \( \mathbf{A} \) is a unit vector along the axis of rotation and \( \mathbf{B} \) is a unit vector along the resultant force on a particle \( P \) of the body, we need to consider the dynamics of a rotating body and the forces acting on it.
Understanding the Setup
In a uniformly rotating body, each particle experiences a centripetal force directed towards the axis of rotation. This force is necessary to keep the particle moving in a circular path. The resultant force on the particle \( P \) can be expressed as the combination of the centripetal force and any other forces acting on it, such as gravitational force if applicable.
Defining the Vectors
Let's break down the vectors:
- Unit Vector \( \mathbf{A} \): This vector points along the axis of rotation. Since it is a unit vector, its magnitude is 1.
- Unit Vector \( \mathbf{B} \): This vector points in the direction of the resultant force on particle \( P \). In the case of uniform rotation, the resultant force is primarily centripetal, directed radially inward toward the axis of rotation.
Analyzing the Dot Product
The dot product \( \mathbf{A} \cdot \mathbf{B} \) gives us a measure of how aligned the two vectors are. Mathematically, the dot product is defined as:
\( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) \)
where \( \theta \) is the angle between the two vectors. Since both \( \mathbf{A} \) and \( \mathbf{B} \) are unit vectors, their magnitudes are 1. Thus, the equation simplifies to:
\( \mathbf{A} \cdot \mathbf{B} = \cos(\theta) \)
Determining the Angle
In the context of a particle in uniform rotation, the direction of the resultant force \( \mathbf{B} \) (which is centripetal) is always directed towards the axis of rotation. Consequently, the angle \( \theta \) between \( \mathbf{A} \) and \( \mathbf{B} \) is 180 degrees, because \( \mathbf{B} \) points inward while \( \mathbf{A} \) points outward along the axis.
Final Calculation
Since \( \theta = 180^\circ \), we have:
\( \cos(180^\circ) = -1 \)
Therefore, the value of the dot product \( \mathbf{A} \cdot \mathbf{B} \) is:
\( \mathbf{A} \cdot \mathbf{B} = -1 \)
Conclusion
In summary, when analyzing the forces acting on a particle in a uniformly rotating body, the dot product of the unit vector along the axis of rotation and the unit vector along the resultant force is -1. This indicates that the two vectors are in opposite directions, reflecting the nature of the centripetal force acting on the particle.
