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

Answer: The torque created by the force vector 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 \(\vec{F} = (2\hat{x} + 3\hat{y})N\) and the position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})m.\) We will use these to find the torque vector \(\vec{\tau}.\)

Use the cross product formula to find the torque vector

The formula to find the torque vector \(\vec{\tau}\) is \(\vec{\tau} = \vec{r} \times \vec{F}\). To find the cross product of two vectors, we can use the determinant method. For \(\vec{r} \times \vec{F}\), it looks like this: $\vec{\tau} = \begin{vmatrix} \hat x & \hat y & \hat z \\ 4 & 4 & 4 \\ 2 & 3 & 0 \end{vmatrix}$

Compute the cross product by expanding the determinant

To calculate the cross product, expand the determinant along the first row: \(\vec{\tau} = \hat{x}\begin{vmatrix} 4 & 4 \\ 3 & 0 \end{vmatrix} - \hat{y}\begin{vmatrix} 4 & 4 \\ 2 & 0 \end{vmatrix} + \hat{z}\begin{vmatrix} 4 & 4 \\ 2 & 3 \end{vmatrix}\) Now, compute the determinants: \(\vec{\tau} = \hat{x}(4 \cdot 0 - 4 \cdot 3) - \hat{y}(4 \cdot 0 - 4 \cdot 2) + \hat{z}(4 \cdot 3 - 4 \cdot 2)\) Simplify the expressions: \(\vec{\tau} = \hat{x}(-12) - \hat{y}(-8) + \hat{z}(4)\)

Write the torque vector in component form

Now that we have calculated the components of the torque vector, we can write it in component form: \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)

Interpret the result

The torque created by the force \(\vec{F} = (2\hat{x} + 3\hat{y})\,\text{N}\) about the pivot point with position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\) is \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\). This means that the force creates a torque that tends to rotate the object in the positive or negative direction along each axis, depending on the sign of the component.

Key concepts

Cross Product

Imagine you have two arrows in space, each pointing in different directions; these arrows can be mathematically described as vectors. The cross product is a special operation that allows us to find another vector that is perpendicular to both of these original vectors. It's like finding a new direction that's completely different from the ones we started with.

In physics, the cross product is particularly useful when calculating rotational effects, such as torque. The result of a cross product, in terms of its magnitude, also represents the area of a parallelogram that the original vectors span. This brings us to torque calculation, where the cross product of a force vector and a position vector yields a torque vector that describes how strongly and in what direction an object will rotate.

Determinant Method

Now, the question arises: How do we actually calculate the cross product of two vectors? That's where the determinant method comes in handy. Picture a square grid with three columns and three rows. By filling in the top row with unit vectors, and the other two with the coordinates of our vectors, we have set up a determinant.

From there, we calculate the determinant by 'cross-multiplying' the entries in a specific way, which is essentially expanding the determinant. For the torque problem, the method involves subtracting the product of certain pairs of numbers and adds the product of others, as determined by the original matrix setup. It’s a systematic way to ensure that we get that perpendicular vector we’ve been looking for.

Forces in Physics

Forces are push or pull interactions that cause objects to accelerate. They’re the bread and butter of Newton’s laws of motion, which govern the way objects move and interact in our universe. When a force is applied to an object at some distance from a pivot point, it can cause the object to rotate; this is called a torque. The further away from the pivot the force is applied, typically, the greater the torque. Remember, in physics, forces are vector quantities — they have both magnitude and direction, which is crucial when calculating things like torque where direction is a fundamental component of the resulting effect.

Vectors in Physics

Vectors provide us a way to represent quantities that have both magnitude and direction, such as forces. They are depicted by arrows where the length represents the magnitude and the arrowhead points in the direction of the vector. Vectors can be added together, or multiplied by a scalar to change their magnitude. A deeper understanding of vectors is essential when you want to predict and describe the physical world, especially in fields like mechanics and electromagnetism. In the context of torque calculation, it’s the vectors representing force and position that we’re interested in, and knowing their interplay is key to solving physical problems.