91Ó°ÊÓ

The cross product is a fundamental mathematical operation in physics used to determine the torque in this exercise. It deals with multiplying two vectors in three-dimensional space to produce another vector that is perpendicular to the plane containing the original vectors.
In physics problems involving forces, the cross product is essential because torque is defined as the cross product of the position vector \( \vec{r} \) and force vector \( \vec{F} \). This relationship can be expressed as:

• Torque \( \vec{\tau} = \vec{r} \times \vec{F} \)

The cross product operation considers the angles between the vectors and calculates the magnitude of the perpendicular component, resulting in a new vector whose direction is determined by the right-hand rule. This operation is vital for calculating the effect of forces acting on objects in rotational motion.

In the context of torque calculations, the position vector \( \vec{r} \) points from a specific point, often the origin, to the point where the force is applied. In this exercise, the plum's coordinates are \((-2.0 \mathrm{~m}, 0, 4.0 \mathrm{~m})\), meaning its position vector \( \vec{r} \) is expressed as:

• \( \vec{r} = -2.0 \hat{i} + 0 \hat{j} + 4.0 \hat{k} \)

Utilizing the position vector is crucial in determining torque because it establishes the lever arm distance from the axis of rotation to the line of action of the force. The longer or shorter the position vector, the larger or smaller the effect of the applied force will be on the rotational motion of the object in question.
Position vectors in problems like this help identify the spatial relationship between points of interest and provide a basis for calculating forces and motions.

In this exercise, the force vector \( \vec{F} \) is explored with different components, each affecting the torque applied to the plum differently. A force vector contains both the magnitude and the direction of force and can be represented in unit-vector notation. In different scenarios:

• If the force is \( F_x = 6.0 \mathrm{~N} \), the vector is \( \vec{F} = 6.0 \hat{i} \)
• For \( F_x = -6.0 \mathrm{~N} \), the vector becomes \( \vec{F} = -6.0 \hat{i} \)
• With \( F_z = 6.0 \mathrm{~N} \), it transforms into \( \vec{F} = 6.0 \hat{k} \)
• And for \( F_z = -6.0 \mathrm{~N} \), it is \( \vec{F} = -6.0 \hat{k} \)

The orientation and magnitude of the force vector directly affect the resulting torque when evaluated via the cross product with the position vector. Depending on their alignment, forces can amplify, decrease, or nullify the torque produced.

Unit-vector notation is a vector expression format that breaks down a vector into its components along the designated coordinate axes. It uses standard unit vectors \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) which point in the x, y, and z directions, respectively.
By using unit-vector notation, you can convey the direction and magnitude of vectors such as the plum’s position vector and various force vectors in this exercise. For example, the position vector \( \vec{r} \) of the plum is:

• \(-2.0 \hat{i} + 0 \hat{j} + 4.0 \hat{k} \)

This notation simplifies both expressing and performing vector operations such as addition, subtraction, and cross products. It’s highly beneficial in a wide range of applications, from simple force systems to complex kinematic analyses, providing a clear and concise representation of vector quantities.