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

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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

01

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 \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\).
02

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: \(\vec{\tau} = \vec{r} \times \vec{F} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ 4 & 4 & 4 \\ 2 & 3 & 0 \end{vmatrix}\)
03

Evaluate the cross product

Now, let's evaluate the cross product using the formula: \(\vec{\tau} = (\hat{x}(4\cdot0 - 3\cdot4) - \hat{y}(4\cdot0 - 2\cdot4) + \hat{z}(4\cdot3 - 2\cdot4))\) \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)
04

Write the final answer

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

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