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You are a mechanical engineer working for a manufacturing
Chapter 1, Problem 1.88(choose chapter or problem)
You are a mechanical engineer working for a manufacturing company. Two forces, \(\vec{F}_{1}\) and \(\vec{F}_{2}\) , act on a component part of a piece of equipment. Your boss asked you to find the magnitude of the larger of these two forces. You can vary the angle between \(\vec{F}_{1}\) and \(\vec{F}_{2}\) from \(0^\circ\) to \(90^\circ\) while the magnitude of each force stays constant. And, you can measure the magnitude of the resultant force they produce (their vector sum), but you cannot directly measure the magnitude of each separate force. You measure the magnitude of the resultant force for four angles \(\theta\) between the directions of the two forces as follows:
\(\begin{array}{lc} \boldsymbol{\theta} & \text { Resultant force }(\mathbf{N}) \\ \hline 0.0^{\circ} & 8.00 \\ 45.0^{\circ} & 7.43 \\ 60.0^{\circ} & 7.00 \\ 90.0^{\circ} & 5.83 \end{array}\)
(a) What is the magnitude of the larger of the two forces?
(b) When the equipment is used on the production line, the angle between the two forces is \(30.0^\circ\). What is the magnitude of the resultant force in this case?
Questions & Answers
QUESTION:
You are a mechanical engineer working for a manufacturing company. Two forces, \(\vec{F}_{1}\) and \(\vec{F}_{2}\) , act on a component part of a piece of equipment. Your boss asked you to find the magnitude of the larger of these two forces. You can vary the angle between \(\vec{F}_{1}\) and \(\vec{F}_{2}\) from \(0^\circ\) to \(90^\circ\) while the magnitude of each force stays constant. And, you can measure the magnitude of the resultant force they produce (their vector sum), but you cannot directly measure the magnitude of each separate force. You measure the magnitude of the resultant force for four angles \(\theta\) between the directions of the two forces as follows:
\(\begin{array}{lc} \boldsymbol{\theta} & \text { Resultant force }(\mathbf{N}) \\ \hline 0.0^{\circ} & 8.00 \\ 45.0^{\circ} & 7.43 \\ 60.0^{\circ} & 7.00 \\ 90.0^{\circ} & 5.83 \end{array}\)
(a) What is the magnitude of the larger of the two forces?
(b) When the equipment is used on the production line, the angle between the two forces is \(30.0^\circ\). What is the magnitude of the resultant force in this case?
ANSWER:Step 1 of 3
Part (a)
The equation for the resultant of two forces at 0° is as follows:
\(R=\sqrt{\left(F_{1}\right)^{2}+\left(F_{2}\right)^{2}+2 F_{1} F_{2} \cos \theta}\)
For \(\theta=0^{\circ}\) and \(R=8.00\ N\) .
\(8=\sqrt{\left(F_{1}\right)^{2}+\left(F_{2}\right)^{2}+2 F_{1} F_{2} \cos 0^{\circ}}\)
\(8=\sqrt{\left(F_{1}\right)^{2}+\left(F_{2}\right)^{2}+2 F_{1} F_{2}(1)}\)
\(8=\sqrt{\left(F_{1}+F_{2}\right)^{2}}\)
\(8=\left(F_{1}+F_{2}\right)\)
\(F_{2}=\left(8-F_{1}\right) \mathrm{N}\)