Question

A 100 N force makes an angle of e with the x-axis
and has a y-component of 30 N. Then the
x-component of the force is
(1) 70 N
(2) 95.4 N
(3) 104.4 N
(4) Zero N​

Answer 1

$\huge\underline\mathfrak\red{Answer}$

WHAT DO YOU MEAN BY COMPONENT OF A VECTOR?

WHEN WE TALK ABOUT COMPONENT OF A VECTOR THAN WE ACTUALLY TALKING ABOUT THE EFFECT OF THAT PARTICULAR VECTOR AT DIFFERENT ANGLES.

$let \:a \: vector \:of \: magnitude \: |a| \ \\ than \: its \: effect \: at \: angle \: \beta \: will \\ be \: |a| \cos \beta$

SOLUTION

$let \: the \: angle \: with \: y \: axis \: is \: \alpha \\ \\ magnitude \: of \: vector = 100 \: newton \\ \\ y \: component = 30 \: newton \\ \\ acc \: to \: discussion \\ \\ 100 \cos \alpha = 30 \\ \\ \cos \alpha = \frac{3}{10} .....(1) \\ \\ by \: squaring \\ \\ {cos}^{2} \alpha = \frac{9}{100} \\ \\ 1 - {sin }^{2} \alpha = \frac{9}{100} \\ \\ 1 - \frac{9}{100} = {sin}^{2} \alpha \\ \\ from \: here \: we \: get \\ \\ \sin \alpha = \frac{ \sqrt{91} }{10} \\ \\ x \: component \: of \: same \: force \\ \\ 100 \cos(90 - \alpha ) \\ \\ where \: 90 - \alpha \: is \: the \: angle \: with \: \\ x \: axis \\ \\ 100 \sin \alpha = 100 \times \frac{ \sqrt{91} }{10} \\ \\ 10 \sqrt{91} = 95.39 \: newton$

Answer 2

Correct Question:

A 100 N force makes an angle of Ф with the x-axis  and has a y-component of 30 N. Then the  x-component of the force is:

Solution:

Force acting in the form of 'x' and 'y' co-ordinates can be divided into two components namely the x-component and the y-component.

The x-component is generally denoted as F CosФ and the y-component is denoted as F SinФ.

According to the question, the y-component has a magnitude of 30 N force. This means that,

⇒ F SinФ = 30 N

⇒ 100 SinФ = 30 N

⇒ Sin Ф = 30 / 100 = 0.3

⇒ Ф is approximately 17°

Now x-component of the force is given as F CosФ

⇒ x-component = 100 × Cos ( 17° )

⇒ x-component = 100 × 0.9536

⇒ x-component = 95.36

Therefore the x-component of the force is 95.36 which is approximately 95.4 N. Hence Option (2) is the right answer.