91Ó°ÊÓ

DATA You are a mechanical engineer working for a manufacturing company. Two forces, \({\boldsymbol{F}}_{1}\) and \({\boldsymbol{F}}_{2}\), 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: $$ \begin{array}{cc} \hline \boldsymbol{\theta} & \text { Resultant force (N) } \\ \hline 0.0^{\circ} & 8.00 \\ 45.0^{\circ} & 7.43 \\ 60.0^{\circ} & 7.00 \\ 90.0^{\circ} & 5.83 \end{array} $$ (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?

Step by step solution

01

Applying Vector Addition Formula

We know that the resultant of two vectors \({\boldsymbol{F}}_{1}\) and \({\boldsymbol{F}}_{2}\), with angle \(\theta\) between them, is given by the formula: \[ R = \sqrt{{F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2}\cos{\theta}} \] where \(R\) is the resultant force. Now, from the data in the table, we know that when the angle \(\theta\) is \(0^{\circ}\), the resultant force \(R\) is \(8.00N\). So, we apply the formula and get the following equation : \[ (8.00)^{2} = {F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2}\cos{0^{\circ}} \] \[ 64 = {F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2} \]

02

Calculate based on Angle at 90 degrees

Next, we know that when \(\theta = 90^{\circ}\), the resultant force \(R\) is \(5.83N\). Applying the same formula, we get the following equation : \[ (5.83)^{2} = {F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2}\cos{90^{\circ}} \] \[ 34 = {F_{1}}^{2} + {F_{2}}^{2} \] We now have two equations involving two unknowns \({F_{1}}^{2}\) and \({F_{2}}^{2}\). We can solve these equations to find each of those variables.

03

Solving Simultaneous Equations

Solving the simultaneous equations obtained in Steps 1 and 2 yields: \[ {F_{1}}^{2} + {F_{2}}^{2} = 34 \] and \[ {F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2} = 64 \] Subtracting these two equations, we find that \[ 2F_{1}F_{2} = 30 \] which gives \(F_{1}F_{2} = 15N^{2}\). Since both forces are equal, \(F_{1} = F_{2}\), so by substituting into the above equation we get \(2F_{1}^{2} = 30N^{2}\), or \(F_{1} = \sqrt{(15)}N = 3.87N\).

04

Calculate Resultant Force at 30 degrees

To solve part (b), that is to find the magnitude of the resultant force when the angle between the two forces is \(30.0^{\circ}\), we again apply the vector addition formula again. \[ R = \sqrt{{F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2}\cos{30^{\circ}}} \] Substituting the value of \(F_{1}\) and \(F_{2}\), we get \[ R = \sqrt{(3.87)^{2} + (3.87)^{2} + 2*3.87*3.87*\cos{30^{\circ}}} \]\[ R = 7.23N \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resultant Force

Understanding the resultant force is crucial when dealing with multiple forces acting on a single point or object. The resultant force is essentially the single force that could replace multiple forces and produce the same effect on the object as the original forces.This is particularly useful in engineering and physics when simplifying complex force systems. To find the resultant force when two individual forces are acting on an object, you use vector addition.

• Vector addition allows you to combine the forces by taking into account both magnitude and direction.
• The mathematical representation involves using the Pythagorean theorem for perpendicular vectors or a more complex equation when vectors form angles other than 90°.

In this exercise, when two forces act at an angle \(\theta\), the resultant force is given by the equation: \[ R = \sqrt{{F_{1}}^{2} + {F_{2}}^{2} + 2F_{1}F_{2}\cos{\theta}} \]This equation shows that the resultant force depends not only on the magnitudes of the forces but also on the angle between them. This means changing the angle directly alters the resultant force.

Vector Components

Vector components are the building blocks of vectors, breaking them into accessible parts typically aligned with the coordinate axes. When dealing with forces, decomposing them into components simplifies the calculation of net forces, especially when forces are not aligned.

• Every vector can be broken down into two perpendicular components, for example, horizontal and vertical components.
• Understanding vector components aids in calculating the resultant force or when integrating forces into more complex systems.

In scenarios such as the original exercise, the specific decomposition of two force vectors simplifies their combination using trigonometry. Consider two forces $F_1$ and $F_2$:

• Each force may have both horizontal and vertical components.
• This helps in performing vector addition efficiently, especially when forces act at various angles.

By understanding vector components, you gain flexibility and precision in your calculations, allowing you to find accurate resultant forces even with complex force interactions.

Force Magnitude Calculation

The magnitude of a force vector gives a measure of the strength of that force. Calculating the force magnitude from components or resultant vectors is a common task in physics and engineering.

• Force magnitude is a scalar quantity, meaning it does not include direction.
• It provides the force's size, telling us how strong the force is without indicating its application direction.

In the exercise, one aims to determine the magnitude of distinct forces when only the resultant force vector is measurable over different angles. Given the aspect where direct measurement of separate forces isn't possible, we rely on simultaneous equations derived from known resultant forces:

• Effective measurement involves formulating equations from known resultant values and incorporating trigonometric analysis.
• Allowing the isolation and identification of unknown force magnitudes.

These calculations help derive individual force magnitudes, paving the way for understanding how each contributes to the total effect on the system.