Question

Resolve F₁ into components along the u and v axes, and determine the magnitudes of these components.

Answer 1

Draw the free-body diagram of the given figure as follows:

Draw the Force triangle with the force \(F_{1}\) along the directions \(u\) and \(v\) as forces \(F_{u}\) and \(F_{v}\) respectively as follows:

Apply the rule of sines to obtain the following relation between the forces:

\(\frac{F_{1}}{\sin 105^{\circ}}=\frac{F_{u}}{\sin 45^{\circ}}=\frac{F_{v}}{\sin 30^{\circ}} \ldots \ldots(1)\)

From the equation (1), obtain the following relation:

\(\frac{F_{1}}{\sin 105^{\circ}}=\frac{F_{u}}{\sin 45^{\circ}}\)

Substitute \(250 \mathrm{~N}\) for \(F_{1}\).

\(\frac{250}{\sin 105^{\circ}}=\frac{F_{u}}{\sin 45^{\circ}}\)

\(F_{u}=\frac{250 \sin 45^{\circ}}{\sin 105^{\circ}}\)

\(F_{u}=183.01 \mathrm{~N}\)

From the equation (1), obtain the following relation:

\(\frac{F_{1}}{\sin 105^{\circ}}=\frac{F_{v}}{\sin 30^{\circ}}\)

Substitute \(250 \mathrm{~N}\) for \(F_{1}\).

\(\frac{250}{\sin 105^{\circ}}=\frac{F_{v}}{\sin 30^{\circ}}\)

\(F_{v}=\frac{250 \times \sin 30^{\circ}}{\sin 105^{\circ}}\)

\(F_{v}=129.41 \mathrm{~N}\)

Therefore, the forces along the \(u\) and \(v\) axes are \({183.01 \mathrm{~N}}\) and \(129.4 \mathrm{~N}\) respectively.