Question

A force, \(\vec{F}=(2 \hat{x}+3 \hat{y}) \mathrm{N},\) is applied to an object at a point whose position vector with respect to the pivot point is \(\vec{r}=(4 \hat{x}+4 \hat{y}+4 \hat{z}) \mathrm{m} .\) Calculate the torque created by the force about that pivot point.

Short answer

Question: Calculate the torque created by the force \(\vec{F} = (2\hat{x} + 3\hat{y})\,\text{N}\) applied at a point with the position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\) with respect to the pivot point. Answer: The torque is \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\).

Step-by-step solution

Write down the given vectors

We are given the force vector \(\vec{F} = (2\hat{x} + 3\hat{y})\,\text{N}\) and the position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\).

Calculate the cross product of \(\vec{r}\) and \(\vec{F}\)

To find the torque vector \(\vec{\tau}\), we need to calculate the cross product of \(\vec{r}\) and \(\vec{F}\). The cross product of two vectors is given by: \(\vec{\tau} = \vec{r} \times \vec{F} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ 4 & 4 & 4 \\ 2 & 3 & 0 \end{vmatrix}\)

Evaluate the cross product

Now, let's evaluate the cross product using the formula: \(\vec{\tau} = (\hat{x}(4\cdot0 - 3\cdot4) - \hat{y}(4\cdot0 - 2\cdot4) + \hat{z}(4\cdot3 - 2\cdot4))\) \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)

Write the final answer

The torque created by the force \(\vec{F}\), about the pivot point with the position vector \(\vec{r}\), is: \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)

Key concepts

Cross Product

In physics and mathematics, the cross product is an operation performed on two vectors. It results in another vector that is perpendicular to the plane formed by the original vectors. This operation is crucial in the calculation of torque. The formula for the cross product of vectors \( \vec{A} = a_1\hat{x} + a_2\hat{y} + a_3\hat{z} \) and \( \vec{B} = b_1\hat{x} + b_2\hat{y} + b_3\hat{z} \) is given by a determinant:

• \( \vec{A} \times \vec{B} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \)

The cross product is not commutative, meaning that \( \vec{A} \times \vec{B} eq \vec{B} \times \vec{A} \). Instead, \( \vec{A} \times \vec{B} = - (\vec{B} \times \vec{A}) \). This property reflects the right-hand rule used to determine the direction of the resulting vector. The magnitude of the cross product is equal to the product of the magnitudes of the two vectors and the sine of the angle between them.

• \( |\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin\theta \)

This makes the cross product a powerful tool for vectorial calculations in 3D space, especially for torque and rotational dynamics.

Vectors

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. Commonly denoted by symbols like \(\vec{A}\), vectors differ from scalars, which only possess magnitude.

• Example: \(\vec{v} = 3\hat{x} + 4\hat{y} \)

Vectors are often represented in component form, using the unit vectors \(\hat{x}, \hat{y}, \) and \(\hat{z}\) in three-dimensional space, corresponding to the directions along the x, y, and z axes. The magnitude of a vector \(\vec{v} = a\hat{x} + b\hat{y} + c\hat{z}\) can be calculated as:

• \( |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \)

Adding or subtracting vectors involves combining their respective components. Unlike scalars, vectors follow unique rules for multiplication, such as the dot and cross products. The cross product, specifically, results in another vector perpendicular to the initial pair, while the dot product results in a scalar. Vectors are indispensable in many fields including engineering, physics, and mathematics because they provide a compact way of describing physical quantities.

Force

Force is a physical quantity that represents an interaction that changes or tends to change the motion of an object. In vector form, it has both magnitude and direction, making it a perfect candidate for vector representation. The basic unit of force in the International System of Units (SI) is the newton (N).

• Example: A force \(\vec{F} = 2\hat{x} + 3\hat{y}\, \mathrm{N} \)

Newton's second law of motion provides a clear relationship between force, mass, and acceleration: \( \vec{F} = m \cdot \vec{a} \), where \(m\) is the mass of an object and \(\vec{a}\) its acceleration. Since forces can be combined using vector addition, understanding their vector nature helps in solving problems involving multiple forces acting on a body. When discussing torque, a force applied at a certain point with respect to a pivot creates rotational effect, which is influenced by both the magnitude of the force and its distance from the pivot, depicted by the position vector.

Position Vector

The position vector is a vector that describes the position of a point in space relative to a reference point or origin. It is often denoted by \(\vec{r}\) and can be expressed in three-dimensional space as \( \vec{r} = x\hat{x} + y\hat{y} + z\hat{z} \).

• Example: \(\vec{r} = 4\hat{x} + 4\hat{y} + 4\hat{z}\, \mathrm{m} \)

The position vector plays a crucial role in physics and engineering, particularly when calculating torque or angular displacement. Torque, for instance, is the rotational equivalent of linear force, calculated as the cross product between the position vector and the force vector acting on a point.

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

This application demonstrates how a position vector not only locates a point in space but also aids in understanding and analyzing rotational motion. Moreover, the length of a position vector is the straight-line distance from the origin to the point, computed as \(|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\). Understanding position vectors is essential for calculating many other physical phenomena like velocity and acceleration in a coordinate system.