Webwork and MasteringPhysics Homework Help: January 2011

Problem 1.71

You are to program a robotic arm on an assembly line to move in the xy-plane. Its first displacement is $\vec A$ ; its second displacement is $\vec B$ , of magnitude

6.40

and direction 62.6 $^\circ$ measured in the sense from the +x-axis toward the -y-axis. The resultant $\vec C= \vec{A} + \vec{B}$ of the two displacements should also have a magnitude of 6.40

, but a direction 22.5 $^\circ$ measured in the sense from the +x-axis toward the +y-axis.

Part A

Find the component $A_x$ of $\vec A$ .

ANSWER:

2.97
Correct

${\rm cm}$

Part B

Find the component $A_y$ of $\vec A$ .

ANSWER:

8.13
Answer Requested

${\rm cm}$

Part C

Find the magnitude of $\vec A$ .

ANSWER:

8.66
Correct

${\rm cm}$

Part D

Find the direction of $\vec A$ measured in the sense from +x - axis toward the +y - axis.

ANSWER:

69.9
Correct

$^\circ$

Exercise 1.41

A disoriented physics professor drives a distance 3.50

north, then a distance 4.70

west, and then a distance 1.55

south.

Part A

Find the magnitude of the resultant displacement, using the method of components.

ANSWER:

5.09
Correct

${\rm km}$

Part B

Find the direction of the resultant displacement, using the method of components.

ANSWER:

67.5
Correct

$^\circ$ West of North

Exercise 1.37

A rocket fires two engines simultaneously. One produces a thrust of 795

directly forward while the other gives a thrust of 484

at an angle 25.0 $^\circ$ above the forward direction.

Part A

Find the magnitude of the resultant force which these engines exert on the rocket.

ANSWER:

1250
Correct

Part B

Find the deviation of the direction (relative to the forward direction) of the resultant force which these engines exert on the rocket.

ANSWER:

9.41
Correct

$^\circ$

Biking Vectors

A student bikes to school by traveling first

$d_N$ = 0.900 $\rm {miles}$ north, then $d_W$ = 0.400 $\rm {miles}$ west, and finally $d_S$ = 0.200 $\rm {miles}$ south.

Part A

If a bird were to start out from the origin (where the student starts) and fly directly (in a straight line) to the school, what distance d_b

would the bird cover?

Hint A.1	How to approach the problem by finding components
*Hint not displayed*

Hint A.2	Find the y component of
*Hint not displayed*

Hint A.3	Find the x component of
*Hint not displayed*

Hint A.4	Magnitude of a vector
*Hint not displayed*

Express your answer in miles.

ANSWER:

0.806
Correct

$\rm miles$

This direct distance is sometimes called the distance "as the crow flies."

Part B

You will now find the same quantity algebraically, without the need to use much geometry. Take the north direction as the positive y direction and east as positive x. The origin is still where the student starts biking.
Let d_vec_N

be the displacement vector corresponding to the first leg of the student's trip. Express d_vec_N

in component form.

Express your answer as two numbers separated by a comma (e.g., 1.0,2.0). By convention, the x component is written first.

ANSWER:

$({d}_N)_x$ , $({d}_N)_y$ =

0,0.900
Correct

Part C

Similarly, let d_vec_W

be the displacement vector corresponding to the second leg of the student's trip. Express d_vec_W

in component form.

Express your answer as two numbers separated by a comma. Be careful with your signs.

ANSWER:

$({d}_{\rm W})_x$ , $({d}_{\rm W})_y$ =

-0.400,0
Correct

Part D

Finally, let d_vec_S

be the displacement vector corresponding to the last leg of the student's trip. Express d_vec_S

in component form.

Express your answer as two numbers separated by a comma. Be careful with your signs.

ANSWER:

$({d}_{\rm S})_x$ , $({d}_{\rm S})_y$ =

0,-0.200
Correct

Part E

The displacement vector for the bird d_vec_b

can be written as $\vec{d}_{\rm N}+\vec{d}_{\rm W}+\vec{d}_{\rm S}$ (see the figure

). In the space provided, type d_vec_b

in component form.

Hint E.1	How to add vectors algebraically
*Hint not displayed*

Express your answer as two numbers separated by a comma. Be careful with your signs.

ANSWER:

$({d}_{\rm b})_x$ , $({d}_{\rm b})_y$ =

-0.400,0.700
Correct

The magnitude

of a vector with components d_x

and

is given by

$d = \sqrt{ {d_x}^2 + {d_y}^2}$ .

Using this definition, you can check that this approach yields the same value for $d_{\rm b} = |\vec{d}_{\rm b}|$ as the one found earlier. Depending on the individual vectors in a given situation, you can decide whether the geometric or the algebraic approach would be more suitable.