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Chapter 3: Q7P (page 57)

Consider two displacements, one of magnitude $3 m$ and another of magnitude $4 m$ . Show how the displacement vectors may be combined to get a resultant displacement of magnitude (a) $7 m$ , (b) $1 m$ , and (c) $5 m$ .

Short Answer

Expert verified

a) Resultant vector for part a) is $7 m$

b) Resultant vector for part a) is $1 m$

c) Resultant vector for part a) is $5 m$

Step by step solution

01

To understand the concept

The vectors can be drawn in different directions and using the vector addition method, the resultant vectors for given vectors can be calculated.

Here the resultant vector is shown by,

$\overset{鈬赌}{R} = \overset{鈬赌}{A} + \overset{鈬赌}{B} (i) R = \sqrt{A^{2} + B^{2}} (ii)$

Where, $\overset{鈬赌}{A}$ and $\overset{鈬赌}{B}$ are the components which is given as

$\overset{鈬赌}{A} = 3 M and \overset{鈬赌}{B} = 4 m$

02

To represent resultant displacement vector of magnitude7 m

The vectors are in the same direction, so using equation (i), the resultant can be written as,

$R_{1} = A + B = 3 + 4 = 7 m$

03

To represent resultant displacement vector of magnitude1m

$R_{2} = (- 3) + 4 = 1 m$

04

To represent resultant displacement vector of magnitude5 m

Using equation (ii) the resultant can be written as

$R_{3} = \sqrt{B^{2} + A^{2}} = \sqrt{4^{2} + 3^{2}} = \sqrt{25} = 5 m$

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Most popular questions from this chapter

Here are three displacements, each measured in meters: ${\overset{鈫�}{d}}_{1} = 4.0 \overset{铃�}{i} + 5.0 \overset{铃�}{j} - 6.0 \overset{铃�}{k}, {\overset{鈫�}{d}}_{2} = - 1.0 \overset{铃�}{i} + 2.0 \overset{铃�}{j} + 3.0 \overset{铃�}{k}$ ,and ${\overset{鈫�}{d}}_{3} = 4.0 \overset{铃�}{i} + 3.0 \overset{铃�}{j} + 2.0 \overset{铃�}{k}$ . (a) What is $\overset{鈫�}{r} = {\overset{鈫�}{d}}_{1} + {\overset{鈫�}{d}}_{2} + {\overset{鈫�}{d}}_{3}$ ? (b) What is the angle between $\overset{鈫�}{r}$ and the positive z axis? (c) What is the component of $d_{1}$ along the direction of ${\overset{鈫�}{d}}_{2}$ (d) What is the component of ${\overset{鈫�}{d}}_{1}$ that is perpendicular to the direction of $\overset{鈫�}{d_{2}}$ and in the plane of role="math" localid="1658465314757" ${\overset{鈫�}{d}}_{1}$ and ${\overset{鈫�}{d}}_{2}$ (Hint: For (c), consider Eq 3-20.and Fig, 3-18; for (d), consider Eq.3-24.)

Find (a) 鈥渘orth cross west,鈥� (b) 鈥渄own dot south,鈥� (c) 鈥渆ast cross up,鈥� (d) 鈥渨est dot west,鈥� and (e) 鈥渟outh cross south.鈥� Let each 鈥渧ector鈥� have unit magnitude

If $\overset{鈫�}{B}$ is added to $\overset{鈫�}{A}$ , the result is $6.0 \hat{i} + 1.0 \hat{j}$ . If $\overset{鈫�}{B}$ is subtracted fromrole="math" localid="1656310989938" $\overset{鈫�}{A}$ , the result is $- 4.0 \hat{i} + 7.0 \hat{j}$ . What is the magnitude of role="math" localid="1656311031101" $\overset{鈫�}{A}$ ?

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: $\overset{鈫�}{d_{1}}$ for 0.40 msouthwest (that is, at $45 掳$ from directly south and from directly west), $\overset{鈫�}{d_{2}}$ for 0.50 mdue east, $\overset{鈫�}{d_{3}}$ for 0.60 mat $60 掳$ north of east. Let the positive x direction be east and the positive y direction be north. What are (a) the x component and (b) the y component of $\overset{鈫�}{d_{1}}$ ? Next, what are (c) the x component and (d) the y component of $\overset{鈫�}{d_{2}}$ ? Also, what are (e) the x component and (f) the y component of $\overset{鈫�}{d_{3}}$ ?

What are (g) the x component, (h) the y component, (i) the magnitude, and (j) the direction of the ant鈥檚 net displacement? If the ant is to return directly to the starting point, (k) how far and (l) in what direction should it move?

In Fig. $3 - 27$ , a heavy piece of machinery is raised by sliding it a distance $d = 12.5 m$ along a plank oriented at angle $胃 = 20.0 掳$ to the horizontal. How far is it moved (a) vertically and (b) horizontally?

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