ვექტორების დამატება კომპონენტების გამოყენებით - პრობლემები და გადაწყვეტილებები

ვექტორების დამატება კომპონენტების გამოყენებით - პრობლემები და გადაწყვეტილებები

1. Three vectors as shown in the figure below.

V₁ = 30

V₂ = 30

V₃ = 40

What is the resultant ვექტორები.

ცნობილი:

V₁ = 30, angle between V₁ and x axis = 30^o

V₂ = 30, angle between V₂ and x axis = 30^o

V₃ = 40, angle between V₃ and x axis = 0^o

სასურველი: The resultant vectors

გამოსავალი:

The components of vectors :

V_1x = (V₁)(cos 30^o) = (30)(0.5√3) = 15√3. Positive because this vector component points along the positive x axis (rightward).

V_1y = (V₁)(ცოდვა 30^o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_2x = (V₂)(cos 30^o) = (30)(0.5√3) = -15√3. Negative because this vector component points along the negative x axis (leftward).

V_2y = (V₂)(ცოდვა 30^o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_3x = (V₃)(cos 0^o) = (40)(1) = 40. Positive because this vector component points along the positive x axis (rightward).

V_3y = (V₃)(ცოდვა 0^o) = (40)(0) = 0

The components of the resultant vectors :

V_x = V_1x - ვ_2x + V_3x = 15√3 – 15√3 + 40 = 40

V_y = V_1y + V_2y + V_3y 15 + 15 = 30

The resultant vector :

2. Two forces perpendicular each other, F₁ = 12 ჩრდილოეთი და დასავლეთი₂ = 5 N. What is the resultant of both forces.

ცნობილი:

აიძულებს 1 (F₁) = 12 ნიუტონი

აიძულებს 2 (F₂) = 5 ნიუტონი

სასურველი: The resultant vectors (ΣF)

გამოსავალი:

ΣF² = ვ₁² + F₂² = 12² + 5² 144 + 25 = 169

ΣF = √169 = 13 Newton

3. Three vectors,

V₁ = 30

V₂ = 30

V₃ = 40

Determine the resultant vectors.

ცნობილი:

v₁ = 30, აკეთებს 30^o about the negative x axis

v₂ = 30, აკეთებს 30^o about the positive x axis

v₃ = 40, აკეთებს 0^o about the positive x axis

სასურველი: The resultant vectors

გამოსავალი:

The components of vectors :

v_1x = v₁ cos 30^o = (30)(0.5√3) = -15√3 (Negative because this vector component points along the negative x axis (leftward))

v_1y = v₁ ცოდვა 30^o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward))

v_2x = v₂ cos 30^o = (30)(0.5√3) = 15√3 (Positive because this vector component points along the positive x axis (rightward))

v_2y = v₂ ცოდვა 30^o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward).)

v_3x = v₃ cos 0^o = (40)(1) = 40 (Positive because this vector component points along the positive x axis (rightward).)

v_3y = v₃ ცოდვა 0^o = (40)(0) = 0

The components of the resultant vectors :

v_x = – v_1x + v_2x + v_3x = -15√3 + 15√3 + 40 = 40

v_y = v_1y + v_2y + v_3y 15 + 15 = 30

The resultant vector :

4. What is the resultant of three vectors as shown in figure below :

ცნობილი:

F₁ = 3 Newton, აკეთებს 60^o about the positive x axis

F₂ = 3 Newton, აკეთებს 0^o about the negative x axis

F₃ = 6 Newton, makes 60^o about the negative y axis

სასურველი: The resultant vector

გამოსავალი:

The components of vectors :

F_1x = ვ₁ cos 60^o = (3)(0.5) = 1.5 N (Positive because this vector component points along the positive x axis (rightward))

F_1y = ვ₁ ცოდვა 60^o = (3)(0.5√3) = 1.5√3 N (Positive because this vector component points along the positive y axis (upward))

F_2x = ვ₂ cos 0^o = (3)(1) = -3 N (Negative because this vector component points along the negative x axis (leftward))

F_2y = ვ₂ ცოდვა 0^o = (3)(0) = 0

F_3x = ვ₃ cos 60^o = (6)(0.5) = 3 N (Positive because this vector component points along the positive x axis (rightward))

F_3y = ვ₃ ცოდვა 60^o = (6)(0.5√3) = -3√3 N (Negative because this vector component points along the negative y axis (ქვემოთward))

The components of the resultant vectors :

ΣF_x = ვ_1x - ფ_2x + F_3x = 1.5 ნ – 3 ნ + 3 ნ = 1.5 ნ

ΣF_y = ვ_1y + F_2y - ფ_3y = 1.5√3 N + 0 N – 3√3 N = -1.5√3 N

The resultant vector :

5. Two forces, F₁ = 15 ჩრდილოეთი და დასავლეთი₂ = 9 N. The angle between both vectors is 60°. What is the resultant of vectors.

სასურველი:

ძალა 1 (F₁) = 15 ნიუტონი

აიძულებს 2 (F₂) = 9 ნიუტონი

კუთხე (θ) = 60^o

სასურველი: The resultant vector

გამოსავალი:

6. What is the resultant of three vectors as shown in the figure below?

ცნობილი:

F₁ = 20 Newton, angle between F₁ and x axis = 0

F₂ = 20 Newton, angle between F₂ and x axis = 60

F₃ = 24 Newton, angle between F₃ and x axis = 60

სასურველი: The resultant vector

გამოსავალი:

The components of vectors :

F_1x = (ფ₁)(cos 0) = (20)(1) = 20. Positive because this vector component points along the positive x axis (rightward)

F_1y = (ფ₁)(sin 0) = (20)(0) = 0

F_2x = (ფ₂)(cos 60) = (20)(0.5) = -10. Negative because this vector component points along the negative x axis (leftward)

F_2y = (ფ₂)(sin 60) = (20)(0.5√3) = 10√3. Positive because this vector component points along the positive y axis (upward)

F_3x = (ფ₃)(cos 60) = (24)(0.5) = -12. Negative because this vector component points along the negative x axis (leftward)

F_3y = (ფ₃)(sin 60) = (24)(0.5√3) = -12√3. Negative because this vector component points along the negative y axis (ქვემოთward)

The components of the resultant vectors :

F_x = ვ_1x - ფ_2x - ფ_3x = 20 – 10 – 12 = -2

F_y = ვ_1y + F_2y - ფ_3y = 0 + 10√3 – 12√3 = -2√3

The resultant vector :