Q6CP If F1鈬赌=<300,0,-200>聽and... [FREE SOLUTION]

Chapter 1: Q6CP (page 14)

If $\overset{鈬赌}{F_{1}} = < 300, 0, - 200 >$ and $\overset{鈬赌}{F_{2}} = <150, - 300, 0>$ , calculate the following
If and , calculate the following:(a) $\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} (b) | \overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} | (c) | \overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} | (d) | \overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} | = | \overset{鈬赌}{F_{1}} | + | \overset{鈬赌}{F_{2}} | ? (e) {\overset{鈬赌}{F}}_{1} - \overset{鈬赌}{F_{2}} (f) | \overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}} | (g) | \overset{鈬赌}{F_{1}} | - | \overset{鈬赌}{F_{2}} | (h) I s | \overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}} | = | \overset{鈬赌}{F_{1}} | - | \overset{鈬赌}{F_{2}} | ?$

Short Answer

Expert verified

The answer to each part is:

$(a) \overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} = <450, - 300, - 200> (b) |\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}}| = 576.6 (c) |\overset{鈬赌}{F_{1}}| + |\overset{鈬赌}{F_{2}}| = 695.9 (d) N o t e q u a l (e) \overset{}{\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}} = <150, 300 - 200>} (f) |\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}}| = 390.5 (g) |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}| = 25.1 (h) N o t e q u a l$

Step by step solution

Basic rule of addition or subtraction of vectors

To add or subtract two vectors, add or subtract the corresponding components. This process is just like adding two constant numbers. Only the parts having the same type of components need to be added or subtracted.

Finding the sum of given vectors

(a)

Add the corresponding components of both the force vectors to obtain the sum of the given vectors

$\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} = <300 + 150, 0 - 300, - 200 + 0> = <450, - 300, - 200>$

Therefore, the vector for is $\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}} i s 鈥� <450, - 300, - 200>$ .

Finding the magnitude of the sum of given vectors

(b)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$|\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}}| = \sqrt{{(450)}^{2} + {(- 350)}^{2} + {(- 200)}^{2}} = \sqrt{202500 + 90000 + 40000} = \sqrt{332500} = 576.6 T h e r e f o r e, t h e v a l u e o f |\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}}| i s 576.6$

Finding the value of F1⇀+F2⇀

(c)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$|\overset{鈬赌}{F_{1}}| = \sqrt{{(300)}^{2} + 0^{2} + {(- 200)}^{2}} = \sqrt{90000 + 0 + 40000} = \sqrt{130000} = 360.5$

(c)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$|\overset{鈬赌}{F_{1}}| = \sqrt{{(150)}^{2} + {(- 300)}^{2} - 0^{2}} = \sqrt{22500 + 90000 + 0} = \sqrt{112500} = 335.4$

Take the sum of obtained magnitudes.

$|\overset{鈬赌}{F_{1}}| + |\overset{鈬赌}{F_{2}}| = 360.5 + 335.4 = 695.9 T h e r e f o r e t h e v a l u e o f |\overset{鈬赌}{F_{1}}| + |\overset{鈬赌}{F_{2}}| i s 695.9$

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

(d)

From part (b) $|\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}}| = 576.6 a n d p a r t (c) |\overset{鈬赌}{F_{1}}| + |\overset{鈬赌}{F_{2}}| = 695.9 T h e r e f o r e t h e v a l u e o f |\overset{鈬赌}{F_{1}} + \overset{鈬赌}{F_{2}}| i s n o t e q u a l t o v a l u e o f |\overset{鈬赌}{F_{1}}| + |\overset{鈬赌}{F_{2}}| .$

Finding the difference between given vectors

(e)

Subtract the corresponding components of both the force vectors to obtain the difference between the given vectors.

$\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}} = <300 - 150, 0 + 300, - 200 - 0> = <150, 300, - 200> T h e r e f o r e t h e v e c t o r f o r \overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}} i s <150, 300, - 200>$

Finding the magnitude of the difference between given vectors

(f)

Take the square root of the sum of the square of each component of the vector to obtain its magnitude.

$|\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}}| = \sqrt{{(150)}^{2} {(300)}^{2} + {(- 200)}^{2}} = \sqrt{22500 + 90000 + 40000} = \sqrt{152500} = 390.5 T h e r e f o r e t h e v a l u e o f |\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}}| i s 390.5$

Finding the value of F1⇀-F2⇀

(g)

From part (c), and $|\overset{鈬赌}{F_{1}}| = 360.5 a n d |\overset{鈬赌}{F_{2}}| = 335.4 T a k e t h e d i f f e r e n c e o f o b t a i n e d m a g n i t u d e s . |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}| = 360.5 - 335.4 T h e r e f o r e t h e v a l u e o f |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}| i s 25.1$ .

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

(h)

From part (f) $|\overset{鈬赌}{F_{1}} - \overset{鈬赌}{F_{2}}| = 390.5 a n d p a r t i s (g) |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}| = 25.1 T h e r e f o r e t h e v a l u e o f |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}| i s n o t e q u a l t o t h e v a l u e o f |\overset{鈬赌}{F_{1}}| - |\overset{鈬赌}{F_{2}}|$

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Location	t(s)	Position ( m)
A	0.0	(0,0,0)
B	1.0	(22.3,26.1,0)
C	2.0	(40.1,38.1,0)

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Short Answer

Step by step solution

Basic rule of addition or subtraction of vectors

Finding the sum of given vectors

Finding the magnitude of the sum of given vectors

Finding the value of F1⇀+F2⇀

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

Finding the difference between given vectors

Finding the magnitude of the difference between given vectors

Finding the value of F1⇀-F2⇀

Analyzing F1⇀+F2⇀=F1⇀+F2⇀

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