Four forces are exerted on the object shown in FIGURE P3.45P3.45. (Forces are measured in newtons, abbreviated N\text{N}.) The net force on the object is F⃗net=F⃗1+F⃗2+F⃗3+F⃗4=4.0 i^ N\vec{F}_{\text{net}}=\vec{F}_1+\vec{F}_2+\vec{F}_3+\vec{F}_4=4.0\,\hat{\mathbf{i}}\,\text{N}. What are (a) F⃗3\vec{F}_3 and (b) F⃗4\vec{F}_4? Give your answers in component form.

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

문제 45

3장, 문제 45

Four forces are exerted on the object shown in FIGURE $P 3.45$ . (Forces are measured in newtons, abbreviated $N$ .) The net force on the object is ${\vec{F}}_{net} = {\vec{F}}_{1} + {\vec{F}}_{2} + {\vec{F}}_{3} + {\vec{F}}_{4} = 4.0 \hat{i} N$ . What are (a) $F ⃗ sub 3$ and (b) $F ⃗ sub 4$ ? Give your answers in component form.

검증된 단계별 안내

Step 1: Break down the forces into their components. For Fₐ (7.0 N upward along the y-axis), the x-component is 0 and the y-component is 7.0 N. For Fᵦ (unknown magnitude along the positive x-axis), the x-component is Fᵦ and the y-component is 0. For F𝒸 (unknown magnitude downward along the negative y-axis), the x-component is 0 and the y-component is -F𝒸. For F𝒹 (5.0 N at 30° below the negative x-axis), calculate its components using trigonometry: x-component = -5.0 cos(30°), y-component = -5.0 sin(30°).

Step 2: Write the equation for the net force in the x-direction. The net force in the x-direction is given as 4.0 N. Using the components: Fₙₑₜₓ = Fᵦ + (-5.0 cos(30°)). Solve for Fᵦ.

Step 3: Write the equation for the net force in the y-direction. The net force in the y-direction is 0 N (since the object is not accelerating vertically). Using the components: Fₙₑₜᵧ = 7.0 + (-F𝒸) + (-5.0 sin(30°). Solve for F𝒸.

Step 4: Combine the results to find F₃ and F₄. F₃ corresponds to Fᵦ, which is purely in the x-direction, and F₄ corresponds to F𝒸, which is purely in the y-direction. Express both forces in component form.

Step 5: Verify the solution by ensuring the calculated components satisfy the given net force conditions (Fₙₑₜₓ = 4.0 N and Fₙₑₜᵧ = 0 N).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Net Force

The net force is the vector sum of all individual forces acting on an object. It determines the object's acceleration according to Newton's second law, F_net = ma. In this problem, the net force is given as 4.0 N in the positive x-direction, which means the combined effect of all forces must result in this specific vector.

Vector Components

Forces can be broken down into their components along the x and y axes. This is essential for analyzing forces in two dimensions. Each force can be expressed as F_x and F_y, allowing for easier calculations of the net force by summing the respective components. In this case, we need to find the components of forces F₃ and F₄.

Trigonometric Functions in Force Analysis

Trigonometric functions, such as sine and cosine, are used to resolve forces into their components when angles are involved. For example, if a force is applied at an angle, its x and y components can be calculated using F_x = F * cos(θ) and F_y = F * sin(θ). This is particularly relevant for force F_d, which is at a 30° angle, requiring the use of trigonometry to find its contributions to the net force.