CALC An object moving in the xy-plane is subjected to the force F⃗=(2xy i^+x2 j^) N\vec{F} = (2xy\,\hat{i} + $$x^2$$\,\hat{j})\,\text{N}, where x and y are in m. Is this a conservative force?

Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

문제 65c

10장, 문제 65c

CALC An object moving in the xy-plane is subjected to the force $\vec{F} = (2 x y \hat{i} + x^{2} \hat{j}) N$ , where x and y are in m. Is this a conservative force?

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To determine if the force is conservative, we need to check if the curl of the force vector field is zero. The curl of a vector field **F** = (Fₓ î + Fᵧ ĵ + F𝓏 k̂) is given by: ∇ × **F** = (∂F𝓏/∂y - ∂Fᵧ/∂z) î + (∂Fₓ/∂z - ∂F𝓏/∂x) ĵ + (∂Fᵧ/∂x - ∂Fₓ/∂y) k̂.

In this problem, the force vector is **F** = (2xy î + x² ĵ). Since there is no k̂ component (F𝓏 = 0), the curl simplifies to: ∇ × **F** = (∂Fᵧ/∂x - ∂Fₓ/∂y) k̂.

Compute the partial derivatives: ∂Fᵧ/∂x = ∂(x²)/∂x = 2x, and ∂Fₓ/∂y = ∂(2xy)/∂y = 2x.

Compare the results: ∂Fᵧ/∂x = 2x and ∂Fₓ/∂y = 2x. Since these are equal, the curl of the force is zero: ∇ × **F** = 0.

Since the curl of the force is zero, the force is conservative. This means that the work done by the force depends only on the initial and final positions, not on the path taken.

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주요 개념

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

Conservative Forces

A conservative force is one where the work done by the force on an object moving between two points is independent of the path taken. This means that the work done in moving an object around a closed loop is zero. Examples include gravitational and electrostatic forces. To determine if a force is conservative, one can check if it can be expressed as the gradient of a potential energy function.

Vector Fields

A vector field assigns a vector to every point in a space, representing quantities that have both magnitude and direction. In this context, the force F is a vector field defined in the xy-plane, where each point (x, y) has a corresponding force vector. Understanding vector fields is crucial for analyzing forces acting on objects in physics, particularly in determining their behavior under various conditions.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation of the field at a point and is calculated using a specific mathematical operation. For a force to be conservative, its curl must be zero everywhere in the region of interest. In this case, calculating the curl of the given force vector F will help determine if it is conservative, as a non-zero curl indicates the presence of rotational effects.