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Ch 10: Interactions and Potential Energy
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 10: Interactions and Potential Energy문제 61a
10장, 문제 61a

A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x2 − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

검증된 단계별 안내
1
Step 1: Recall that equilibrium positions occur where the force acting on the particle is zero. The force is related to the potential energy by the equation F(x) = -dU(x)/dx. Begin by calculating the derivative of the given potential energy function U(x) = A/x^2 - B/x with respect to x.
Step 2: Differentiate U(x) with respect to x. Using the power rule for derivatives, the derivative of A/x^2 is -2A/x^3, and the derivative of -B/x is B/x^2. Combine these results to find dU(x)/dx = -2A/x^3 + B/x^2.
Step 3: Set the force equal to zero to find the equilibrium positions. This means solving the equation -dU(x)/dx = 0, or equivalently, -2A/x^3 + B/x^2 = 0.
Step 4: Factorize the equation to simplify. Rewrite the equation as (B/x^2) - (2A/x^3) = 0. Factor out 1/x^2, resulting in (1/x^2)(B - 2A/x) = 0. Since 1/x^2 cannot be zero, solve the remaining equation B - 2A/x = 0.
Step 5: Solve for x to find the equilibrium positions. Rearrange the equation B - 2A/x = 0 to isolate x, giving x = 2A/B. This is the position where the particle is in equilibrium.

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주요 개념

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

Potential Energy

Potential energy is the energy stored in a system due to its position or configuration. In this case, the potential energy U(x) is defined as a function of the particle's position along the x-axis. Understanding how potential energy varies with position is crucial for analyzing the forces acting on the particle and determining its equilibrium positions.
추천 영상:
가이드 코스
07:24
Potential Energy Graphs

Equilibrium Position

An equilibrium position occurs when the net force acting on a particle is zero, meaning the particle is in a state of balance. For a particle moving along the x-axis, this can be found by setting the derivative of the potential energy function U(x) to zero. At these points, the particle experiences no net force and can remain at rest or move with constant velocity.
추천 영상:
가이드 코스
05:50
Forces & Equilibrium Positions

Force from Potential Energy

The force acting on a particle can be derived from the potential energy function using the relationship F(x) = -dU/dx. This means that the force is equal to the negative gradient of the potential energy with respect to position. By calculating this derivative, one can identify the points where the force is zero, which correspond to the equilibrium positions of the particle.
추천 영상:
가이드 코스
07:24
Potential Energy Graphs
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