001 Horizontal and vertical componets of planar forces

Problem 001
Problem Determine the x and y components of the forces shown below in Fig P-001.

Solution 001

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$F_{x1} = 58 \cos 30^\circ = 50.23 \, \text{ kN}$

$F_{y1} = 58 \sin 30^\circ = 29 \, \text{ kN}$

$F_{x2} = -50 \cos 45^\circ = -35.36 \, \text{ kN}$

$F_{y2} = 50 \sin 45^\circ = 35.36 \, \text{ kN}$

Solution of components of forces in space

$F_{x3} = -45(\frac{5}{13}) = -17.31 \, \text{ kN}$

$F_{y3} = -45(\frac{12}{13}) = -41.54 \, \text{ kN}$

$F_{x4} = 40 \, \text{ kN}$

$F_{y4} = 0$

Rectangular Representation
${\bf F} = F (\cos \theta_x {\bf i} + \sin \theta_x {\bf j})$

${\bf F_1} = 58 (\cos 30^\circ {\bf i} + \sin 30^\circ {\bf j}) = 50.23 {\bf i} + 29 {\bf j} \, \text{ kN}$

${\bf F_2} = 50 (-\cos 45^\circ {\bf i} + \sin 45^\circ {\bf j}) = -35.36 {\bf i} + 35.36 {\bf j} \, \text{ kN}$

${\bf F_3} = 45 (-\frac{5}{13} {\bf i} - \frac{12}{13} {\bf j}) = -17.31 {\bf i} - 41.54 {\bf j} \, \text{ kN}$

${\bf F_4} = 40 {\bf i} \, \text{ kN}$

From the above vector notations, F_x is the coefficient of i and F_y is the coefficient of j.

Calculator Techniques

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The calculator keys here are based on the following CASIO models: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus

Technique 1 - Calculator in CMPLX mode: [MODE] → 2:CMPLX

For F₁ = 58 kN:

58 ∠ 30 = $29\sqrt{3} + 29{\bf i}$

To get the decimal display, input [S⇔D]
58 ∠ 30 = $50.23 + 29{\bf i}$

Thus,
$F_{x1} = 50.23 \, \text{kN}$

$F_{y1} = 29 \, \text{kN}$

Note: To enter the ∠ in the operation, input [SHIFT] → [(–)] and the calculator will display the symbol ∠. If not, that means, you are not yet in CMPLX mode.

For F₂ = 50 kN:

50 ∠ (180 - 45) = $-25\sqrt{2} + 25\sqrt{2}{\bf i}$

Press [S⇔D] $-35.36 + 35.36{\bf i}$

Thus,
$F_{x2} = -35.36 \, \text{kN}$

$F_{y2} = 35.36 \, \text{kN}$

For F₃ = 45 kN:

45 ∠ (180 + tan^-1(12/5)) = $-\frac{225}{13} - \frac{540}{13}{\bf i}$

Press [S⇔D] $-17.31 - 41.54{\bf i}$

Thus,
$F_{x3} = -17.31 \, \text{kN}$

$F_{y3} = -41.54 \, \text{kN}$

Technique 2 - Using Rec. Calculator in COMP mode: [MODE] → 1:COMP
To enter Rec input [SHIFT] → [–] and for comma " , " input [SHIFT] → [ ) ]

Note: For this method, the answer in X and Y are automatically stored to variable X and Y, replacing any stored values in those variables. To recall the new values, simply input [ALPHA] X or [ALPHA] Y

For F₁ = 58 kN:

Rec(58,30)=
X = 50.23, Y = 29

For F₂ = 50 kN:

Rec(50,(180-45))=
X = -35.36, Y = 35.36

For F₃ = 45 kN:

Rec(45,(180+tan^-1(12/5))=
X = -17.31, Y = -41.54

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