Question 1

As shown in the figure,three force vectors act on an object.The magnitudes of the forces as shown in the figure are F₁ = 80.0 N,F₂ = 60.0 N,and F₃ = 40.0 N,where N is the standard SI unit of force.The resultant force acting on the object is given by A) 180 N at an angle 60.0° with respect to +x-axis. B) 60.0 N at an angle 90.0° with respect to +x-axis. C) 20.0 N at an angle 34.3° with respect to +x-axis. D) 35.5 N at an angle 34.3° with respect to +x-axis. E) 40.0 N at an angle 60.0° with respect to +x-axis.

Accepted Answer

To find the resultant force, we need to add the three force vectors by vector addition. One way to do this is by using the component method. First, we need to resolve each force vector into its x- and y-components.

F₁ = 80 N @ 120° = (-40 N, 69.3 N)
F₂ = 60 N @ 30° = (51.96 N, 30 N)
F₃ = 40 N @ 270° = (0 N, -40 N)

The @ symbol stands for "at an angle". For example, F₁ = 80 N @ 120° means that the magnitude of F₁ is 80 N and its direction is 120° counterclockwise from the +x-axis.

Now we can add the x- and y-components separately:
Fₓ = -40 N + 51.96 N + 0 N = 11.96 N
Fᵧ = 69.3 N + 30 N - 40 N = 59.3 N

The resultant force Fᵣ is the vector sum of Fₓ and Fᵧ:
|Fᵣ| = √(Fₓ² + Fᵧ²) = √(11.96² + 59.3²) = 60.9 N
θ = tan⁻¹(Fᵧ/Fₓ) = tan⁻¹(59.3/11.96) = 79.3°

However, this is not the final answer because the question asks for the angle with respect to the +x-axis, not with respect to the horizontal. To get the angle with respect to the +x-axis, we need to subtract 90° from the angle we just found:
θ' = 79.3° - 90° = -10.7°

This means that the resultant force is 60.9 N at an angle of 10.7° below the -x-axis. However, the question asks for the angle with respect to the +x-axis, so we need to add 180° (or subtract -180°) to this angle:
θ'' = -10.7° + 180° = 169.3°

So the final answer is 35.5 N at an angle of 169.3° with respect to the +x-axis (which is equivalent to saying at an angle of 10.7° with respect to the -x-axis). Therefore, the correct choice is D.

Question 2

Vector  = -3.00 
+ 3)00 
And vector 
= 3)00 
+ 4)00 
) What is vector 
= 
+ 
?
A) 0.00  + 3.00

B) 7.00  + 7.00

C) -3.00  + 7.00

D) 0.00  + 7.00

E) -3.00  -3.00

Accepted Answer

The answer of Vector  = -3.00 
+ 3)00 
And...

Question 3

A teacher sends her students on a treasure hunt. She gives the following instructions:
1) Walk 300 m north
2) Walk 400 m northwest
3) Walk 700 m east-southeast and the treasure is buried there.
As all the other students walk off following the instructions,Jane physics student quickly adds the displacements and walks in a straight line to find the treasure. How far and in what direction does Jane need to walk?
A) 187 m in a direction 67.3° north of east
B) 481 m in a direction 40.9° north of east
C) 399 m in a direction 52.5° north of east
D) 284 m in a direction 28.2° west of north
E) The treasure position cannot be reached in one straight walk.

Accepted Answer

To find the total displacement of the treasure from the starting point, we need to add the three given displacements.

The first displacement of 300 m north is solely in the y-direction.

For the second displacement of 400 m northwest, we can break it down into its x and y components using the angle of 315 degrees (45 degrees from west):

x = 400 cos(315) = -282.8 m (east is positive, so west is negative)
y = 400 sin(315) = 282.8 m (north is positive)

Note the negative sign on the x-component since we are going west.

For the third displacement of 700 m east-southeast, we need to use the angle of 67.5 degrees from east-south-east:

x = 700 cos(67.5) = 309.8 m (east is positive)
y = -700 sin(67.5) = -615.9 m (north is positive)

Note the negative sign on the y-component since we are going south.

Now we can add up the x and y components:

Total x displacement = 300 - 282.8 + 309.8 = 327.0 m (east is positive) 
Total y displacement = 0 + 282.8 - 615.9 = -333.1 m (north is positive)

The distance Jane needs to walk to get to the treasure is the magnitude of the total displacement:

distance = sqrt((327.0)^2 + (-333.1)^2) = 481.1 m

The direction Jane needs to walk can be found using the inverse tangent function:

tan(theta) = (-333.1) / 327.0 
theta = tan^-1(-333.1 / 327.0) = -40.9 degrees

Note that we have a negative angle because we are going west of north. However, the question is asking for the direction relative to east, so we need to add 90 degrees to this angle:

theta = -40.9 + 90 = 49.1 degrees

Finally, we need to take the complement of this angle (90 - theta) to get the angle relative to north of east:

angle = 90 - 49.1 = 40.9 degrees

So Jane needs to walk 481 m in a direction 40.9 degrees north of east to find the treasure.

Question 4

Determine the angle between the directions of vector  = 3.00 
+ 1)00 
And vector 
= -3.00 
+ 3)00  .
A) 26.6°
B) 30.0°
C) 88.1°
D) 117°
E) 45.2°

Accepted Answer

The answer of Determine the angle between the directions of...

Question 5

The components of vector are A_x = +2.2 and A_y = -6.9,and the components of vector Are given are B_x = -6.1 and B_y = -2.2.What is the magnitude of the vector - ? A) 9.5 B) 6.1 C) 9.9 D) 91 E) 0.76

Accepted Answer

The answer of The components of vector are A_x...

Question 6

The angle between vector  = 2.00 
+ 3)00 
And vector 
Is 45.0°. The scalar product of vectors 
And 
Is 3.00. If the x component of vector 
Is positive,what is vector  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of The angle between vector  = 2.00...

Question 7

The scalar product of vector  = 3.00 
+ 2)00 
And vector 
Is 10.0. Which of the following vectors could be vector 
?
A) 2.00  + 4.00

B) 4.00  + 6.00

C) 5.00  + 4.00

D) 12.0 
E) 2.00  + 2.00

Accepted Answer

The answer of The scalar product of vector  =...

Question 8

An airplane undergoes the following displacements: First,it flies 66 km in a direction 30° east of north.Next,it flies 49 km due south.Finally,it flies 100 km 30° north of west.Using vector components,determine how far the airplane ends up from its starting point.
A) 79 km
B) 81 km
C) 82 km
D) 78 km
E) 76 km

Accepted Answer

To solve this, we break each displacement into its north-south and east-west components and then sum these components separately. 1. For the first displacement: - North component = 66 km * cos(30°) = 57.2 km- East component = 66 km * sin(30°) = 33 km2. The second displacement is directly south, so:- North component = -49 km (since it's south, it's negative)- East component = 0 km3. For the third displacement:- North component = 100 km * cos(30°) = 86.6 km- West component = 100 km * sin(30°) = -50 km (since it's west, it's negative)Summing up the components:- Total north-south = 57.2 km (north) - 49 km (south) + 86.6 km (north) = 94.8 km (north)- Total east-west = 33 km (east) - 50 km (west) = -17 km (west)The total displacement is the hypotenuse of the right triangle formed by these components, which can be found using the Pythagorean theorem:$$ \sqrt{(94.8)^2 + (-17)^2} = \sqrt{8990.24 + 289} = \sqrt{9279.24} \approx 79 \, \text{km} $$

Question 9

In the figure,the magnitude of vector  
is 18.0 units,and the magnitude of vector  
is 12.0 units.What vector  
must be added to the vectors  
and  
so that the resultant of these three vectors points in the -x direction and has a magnitude of 7.50 units? Use vector components to find your answer,and express vector  
by giving its magnitude and the angle it makes with the +x-axis taking counterclockwise to be positive.

Accepted Answer

The answer of In the figure,the magnitude of vector ...

Question 10

Vector  = -1.00 
+ -2.00 
And vector 
= 3)00 
+ 4)00 
) What are the magnitude and direction of vector 
= 3)00 
+ 2)00 
?
A) 3.61 in a direction -56.3° counterclockwise from the positive x-axis
B) 3.61 in a direction 56.3° counterclockwise from the positive x-axis
C) 3.61 in a direction 33.7° counterclockwise from the positive x-axis
D) 5.00 in a direction 56.3° counterclockwise from the positive x axis
E) 6.72 in a direction 34.4° counterclockwise from the positive x-axis

Accepted Answer

The answer of Vector  = -1.00 
+ -2.00 
And...

Question 11

Determine the scalar product of  = 6.0 
+ 4)0 
- 2)0 
And 
= 5)0 
- 6)0 
- 3)0  .
A) 30  + 24
 + 6

B) 30  - 24
 + 6

C) 12
D) 60
E) undefined

Accepted Answer

The answer of Determine the scalar product of  =...

Question 12

A helicopter is flying horizontally with a speed of 444 m/s over a hill that slopes upward with a 2% grade (that is,the "rise" is 2% of the "run").What is the component of the helicopter's velocity perpendicular to the sloping surface of the hill?
A) 8.9 m/s
B) 220 m/s
C) 435 m/s
D) 444 m/s

Accepted Answer

The perpendicular component of the helicopter's velocity is equal to its horizontal velocity multiplied by the sine of the angle of the hill's slope.

The angle of the slope can be found using trigonometry:

tan(theta) = rise/run = 0.02

theta = tan^(-1)(0.02)

theta = 1.143 degrees

The sine of theta is: sin(theta) = 0.02/sqrt(1 + 0.02^2)

sin(theta) ≈ 0.02

So the perpendicular component of the helicopter's velocity is:

perpendicular component = horizontal velocity * sin(theta)

perpendicular component = 444 m/s * 0.02

perpendicular component ≈ 8.9 m/s

Therefore, the answer is A) 8.9 m/s.

Question 13

Vector  = 1.00 
+ -2.00 
And vector 
= 3)00 
+ 4)00 
) What are the magnitude and direction of vector 
= 
+ 
?
A) 7.21 in a direction 33.7° counterclockwise from the positive x axis
B) 6.00 in a direction 63.4° counterclockwise from the positive x axis
C) 4.47 in a direction 6.34° counterclockwise from the positive x axis
D) 4.47 in a direction 26.6° counterclockwise from the positive x axis
E) 7.21 in a direction 56.3° counterclockwise from the positive x axis

Accepted Answer

The magnitude of the vector can be found using the Pythagorean theorem:
√[(3.00)^2 + (4.00)^2] = 5.00
The direction of the vector can be found using trigonometry:
θ = tan⁻¹(4.00/3.00) = 53.1° counterclockwise from the positive x-axis
However, since the vector components have a negative coefficient, the direction will actually be in the opposite direction, which is
θ = 180° - 53.1° = 126.9°, or 360° - 53.1° = 306.9° counterclockwise from the positive x-axis
Rounding to one decimal place gives 26.9°, which is closest to 26.6°, so the answer is (D).

Question 14

The components of vector are B_x = -3.5 and B_y = -9.7,and the components of vector Are C_x = -6 and C_y = +8.1.What is the angle (less than 180 degrees)between vectors And ? A) 124° B) 56° C) 17° D) 163° E) 106°

Accepted Answer

The answer of The components of vector are B_x...

Question 15

Vectors  and 
Are shown in the figure. What is 
? 
A) 31.8
B) -32.0  - 2.00

C) 1028
D) 34.0
E) -2.00  - 32.0

Accepted Answer

We can use the Pythagorean theorem to find the magnitude of the vector 11ea8401_fc14_da06_aeff_31a6f2ff41f5_TB6682_11 which is approximately 40.2 units. Then, we can use the angle shown in the diagram and some trigonometry to find the x-component and y-component of the resultant vector. Adding these components together, we get a value of approximately 31.8 units.

Question 16

If  = +4 
- 2 
- 3 
And 
= -4 
-2 
- 3 
,which of the following numbers is closest to the magnitude of 
- 
?
A) 8
B) 7
C) 9
D) 10
E) 11

Accepted Answer

The answer of If  = +4 
- 2 
-...

Question 17

An apple falls from an apple tree growing on a 20° slope.The apple hits the ground with an impact velocity of 16.2 m/s straight downward.What is the component of the apple's impact velocity parallel to the surface of the slope?
A) 5.5 m/s
B) 8.7 m/s
C) 12 m/s
D) 15 m/s

Accepted Answer

The answer of An apple falls from an apple tree...

Question 18

What is the magnitude of  + 
+ 
,where 
= 1)00 
+ 4)00 
- 1)00  , 
= 3)00 
- 1)00 
- 4)00 
And 
= -1.00 
+ 1)00 
?
A) 7.07
B) 2.00
C) 10.76
D) 6.78
E) 8.12

Accepted Answer

The answer of What is the magnitude of  +...

Question 19

Three forces are exerted on an object placed on a tilted floor.Forces are vectors.The three forces are directed as shown in the figure. If the forces have magnitudes F₁ = 1.0 N,F₂ = 8.0 N and F₃ = 7.0 N, where N is the standard unit of force,what is the component of the net force _net = ₁+ ₂+ ₃ parallel to the floor? A) 2.5 N B) 5.1 N C) 6.0 N D) 7.8 N

Accepted Answer

The answer of Three forces are exerted on an object...

Question 20

Vector  
has a magnitude of 75.0 cm and points at 30° above the positive x-axis. Vector  
has a magnitude of 25.0 cm and points along the negative x-axis. Vector  
has a magnitude of 40.0 cm and points at 45° below the negative x-axis.
(a)Determine the x and y components of Vector  .
(b)Determine the x and y components of Vector  .
(c)Determine the x and y components of Vector  .
(d)Determine x and y components of the sum of these three vectors.
(e)Determine the magnitude and direction of the sum of these three vectors.

Accepted Answer

The answer of Vector  
has a magnitude of 75.0...

Quiz 1: A: Foundations

As shown in the figure,three force vectors act on an object.The magnitudes of the forces as shown in the figure are F₁ = 80.0 N,F₂ = 60.0 N,and F₃ = 40.0 N,where N is the standard SI unit of force.The resultant force acting on the object is given by

Vector = -3.00 + 3) 00 And vector = 3) 00 + 4) 00 ) What is vector = + ?

Determine the angle between the directions of vector = 3.00 + 1) 00 And vector = -3.00 + 3) 00 .

The components of vector are A_x = +2.2 and A_y = -6.9,and the components of vector Are given are B_x = -6.1 and B_y = -2.2.What is the magnitude of the vector - ?

The angle between vector = 2.00 + 3) 00 And vector Is 45.0°. The scalar product of vectors And Is 3.00. If the x component of vector Is positive,what is vector .

The scalar product of vector = 3.00 + 2) 00 And vector Is 10.0. Which of the following vectors could be vector ?

An airplane undergoes the following displacements: First,it flies 66 km in a direction 30° east of north.Next,it flies 49 km due south.Finally,it flies 100 km 30° north of west.Using vector components,determine how far the airplane ends up from its starting point.

Vector = -1.00 + -2.00 And vector = 3) 00 + 4) 00 ) What are the magnitude and direction of vector = 3) 00 + 2) 00 ?

Determine the scalar product of = 6.0 + 4) 0 - 2) 0 And = 5) 0 - 6) 0 - 3) 0 .

A helicopter is flying horizontally with a speed of 444 m/s over a hill that slopes upward with a 2% grade (that is,the "rise" is 2% of the "run") .What is the component of the helicopter's velocity perpendicular to the sloping surface of the hill?

Vector = 1.00 + -2.00 And vector = 3) 00 + 4) 00 ) What are the magnitude and direction of vector = + ?

The components of vector are B_x = -3.5 and B_y = -9.7,and the components of vector Are C_x = -6 and C_y = +8.1.What is the angle (less than 180 degrees) between vectors And ?

Vectors and Are shown in the figure. What is ?

If = +4 - 2 - 3 And = -4 -2 - 3 ,which of the following numbers is closest to the magnitude of - ?

An apple falls from an apple tree growing on a 20° slope.The apple hits the ground with an impact velocity of 16.2 m/s straight downward.What is the component of the apple's impact velocity parallel to the surface of the slope?

What is the magnitude of + + ,where = 1) 00 + 4) 00 - 1) 00 , = 3) 00 - 1) 00 - 4) 00 And = -1.00 + 1) 00 ?