Please help me in a Physics proof!

I got this problem in the book, Elements of Statics and Dynamics. Please help me in the problem! Thanks!

If the resultant of two forces acting on a particle be at right angles to one of them, and its magnitude is one third of the other, show that the ratio of the longer force to the smaller is $3:2\sqrt { 2 }$

#Mechanics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Let us assume that the resultant force is in the horizontal direction and the force $F_1$ in the vertical direction. The direction of $F_2$ will be as shown in the figure. The vertical component of $F_2$ should be equal to $F_1$ since there is no resultant force in vertical direction. It is given that $F_2 = F_r/3$ i.e. the horizontal component of $F_2 = F_2/3$ . By simple trigonometry we get that $\sin \theta = 1/3$ where $\theta$ is the angle between the vertical and $F_2$ . and therefore, $\cos \theta = 2\sqrt{2}/3$ . Therefore, again by trigonometry, $F_2 = F_1 \times 2\sqrt{2}/3$ Therefore, we get the required ratio !! i'm not able to post a figure now, will post it later.

CH Nikhil - 6 years ago

Draw them as a right triangle. Use geometry for it. Take the resultant as $x$ . The other vector will be $3x$ . Now take the ratio.

Aditya Kumar - 6 years ago

Can you make a effort to show the process?

Swapnil Das - 6 years ago

Dude its just pythagoras theorem.

Aditya Kumar - 6 years ago

@Aditya Kumar – Oh, I am getting a physics feeling...

Swapnil Das - 6 years ago

@Aditya Kumar – Thank You brother, I got your method!

Swapnil Das - 6 years ago

@Swapnil Das – Any day!

Aditya Kumar - 6 years ago

If we assume the resultant of the given forces to lie on the co-ordinate axes such that the resultant is along the x-axis, F1 is along the y-axis and F2 is in the third quadrant making an angle A with the x-axis, then this problem is a breeze... component of F2 along y-axis is equal to F1, since the resultant is perpendicular to it,and only the component of F2 along the resultant(x axis), has contributed to the resultant..... HOPE IT HELPS....

Surya Pratap Singh - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$