The board is used to hold


The board is used to hold the end of a four-way lug wrench in position. If a torque of 30 N•m about the x axis is required to tighten the nut, determine the required magnitude of the force F that the man’s foot must apply on the end of the wrench in order to turn it. Force F lies in a vertical plane.

The board is used to hold

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

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Let us first draw a simplified diagram.

The board is used to hold

Notice the position vector, starting from location A to location B. Let us define this position vector in Cartesian vector form. The locations of points A and B are:

A:(0.25i+0j+0k) m

B:(0.25i+0.25j+0k) m

 

Position vector r_{AB} is then:

r_{AB}=\left\{(0.25-0.25)i+(0.25-0)j+(0-0)k\right\}

r_{AB}=\left\{0i+0.25j+0k\right\} m

 

We will now express the force F in Cartesian form. Remember that force F will have two components, a y-component and a z-component.

F=\left\{0i+F\cos60^0j-F\sin60^0k\right\}

(simplify)

F=\left\{0i+0.5F\,j-0.866F\,k\right\}

Note that our z-component is negative because the z-component of the force is going down (or lies in the negative z-axis).

 

We can now find the required force by finding the moment created along the x-axis. This value is given to us in the question as 30N•m. The moment along the x-axis is:

M_x=i\cdot r_{AB}\times F

(Remember that the x-axis has a unit vector of i)

 

Substitute the values we know:

30=\begin{bmatrix}1&0&0\\0&0.25&0\\0&0.5F&-0.866F\end{bmatrix}

(Solve by taking the cross product)

30=1\left[(0.25)(-0.866F)-(0.5F)(0)\right]+0+0

30=-0.2165F

The negative sign indicates that the moment created is towards the negative x-axis. 

30=0.2165F

F=138.57 N

 

Final Answer:

F=138.57 N

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 4, question 4-55.

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