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Conservative Forces and Potential Energy | Study Mondo
Topics / Work and Energy / Conservative Forces and Potential Energy Conservative Forces and Potential Energy Path independence, potential energy functions, and mechanical energy conservation
๐ฏ โญ INTERACTIVE LESSON
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Start Interactive Lesson โ Conservative Forces and Potential Energy
Conservative Force Definition
A force is conservative if the work done is independent of path (depends only on endpoints).
Equivalently:
Work around any closed path is zero: โฎ F โ โ
d r โ = 0 \oint \vec{F} \cdot d\vec{r} = 0 โฎ F โ
๐ Practice ProblemsNo example problems available yet.
Explain using: ๐ Simple words ๐ Analogy ๐จ Visual desc. ๐ Example ๐ก Explain
๐งช Practice Lab Interactive practice problems for Conservative Forces and Potential Energy
โพ ๐ Related Topics in Work and Energyโ Frequently Asked QuestionsWhat is Conservative Forces and Potential Energy?โพ Path independence, potential energy functions, and mechanical energy conservation
How can I study Conservative Forces and Potential Energy effectively?โพ Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Regular review and active practice are key to retention.
Is this Conservative Forces and Potential Energy study guide free?โพ Yes โ all study notes, flashcards, and practice problems for Conservative Forces and Potential Energy on Study Mondo are 100% free. No account is needed to access the content.
What course covers Conservative Forces and Potential Energy?โพ Conservative Forces and Potential Energy is part of the AP Physics C: Mechanics course on Study Mondo, specifically in the Work and Energy section. You can explore the full course for more related topics and practice resources.
๐ก Study Tipsโ Work through examples step-by-step โ Practice with flashcards daily โ Review common mistakes
d
=
0
The force can be written as: F โ = โ โ U \vec{F} = -\nabla U F = โ โ U
Potential Energy For a conservative force:
W = โ ฮ U = โ ( U f โ U i ) W = -\Delta U = -(U_f - U_i) W = โ ฮ U = โ ( U f โ โ U i โ )
Differential form:
d W = F โ โ
d r โ = โ d U dW = \vec{F} \cdot d\vec{r} = -dU d W = F โ
d r = โ d U
Finding U from F In one dimension:
F x = โ d U d x F_x = -\frac{dU}{dx} F x โ = โ d x d U โ
U ( x ) = โ โซ F x โ d x + C U(x) = -\int F_x \, dx + C U ( x ) = โ โซ F x โ d x + C
In three dimensions:
F โ = โ โ U = โ ( โ U โ x i ^ + โ U โ y j ^ + โ U โ z k ^ ) \vec{F} = -\nabla U = -\left(\frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k}\right) F = โ โ U = โ ( โ x โ U โ i ^ + โ y โ U
Finding F from U F x = โ โ U โ x , F y = โ โ U โ y , F z = โ โ U โ z F_x = -\frac{\partial U}{\partial x}, \quad F_y = -\frac{\partial U}{\partial y}, \quad F_z = -\frac{\partial U}{\partial z} F x โ = โ โ x โ U โ , F y โ = โ โ y โ U โ , F z โ = โ โ z โ U โ
Common Potential Energies
Gravitational (near Earth) F y = โ d U d y = โ m g F_y = -\frac{dU}{dy} = -mg F y โ = โ d y d U โ = โ m g
Spring U s = 1 2 k x 2 U_s = \frac{1}{2}kx^2 U s โ = 2 1 โ k x 2
F x = โ d U d x = โ k x F_x = -\frac{dU}{dx} = -kx F x โ = โ d x d U โ = โ k x
Universal Gravitation U g = โ G m 1 m 2 r U_g = -\frac{Gm_1m_2}{r} U g โ = โ r G m 1 โ m 2 โ โ
F r = โ d U d r = โ G m 1 m 2 r 2 F_r = -\frac{dU}{dr} = -\frac{Gm_1m_2}{r^2} F r โ = โ d r d U โ = โ r 2 G m 1 โ m 2 โ โ
(Choosing U = 0 U = 0 U = 0 r = โ r = \infty r = โ
Electric Potential Energy U e = k q 1 q 2 r U_e = k\frac{q_1q_2}{r} U e โ = k r q 1 โ q 2 โ โ
F r = โ d U d r = โ k q 1 q 2 r 2 F_r = -\frac{dU}{dr} = -k\frac{q_1q_2}{r^2} F r โ = โ d r d U โ = โ k r 2 q 1 โ q 2 โ โ
Conservation of Mechanical Energy For conservative forces only:
E = K E + U = constant E = KE + U = \text{constant} E = K E + U = constant
1 2 m v 2 + U ( x ) = E \frac{1}{2}mv^2 + U(x) = E 2 1 โ m v 2 + U ( x ) = E
Taking time derivative:
m v d v d t + d U d t = 0 mv\frac{dv}{dt} + \frac{dU}{dt} = 0 m v d t d v โ + d t d U โ = 0
m v d v d t + d U d x d x d t = 0 mv\frac{dv}{dt} + \frac{dU}{dx}\frac{dx}{dt} = 0 m v d t d v โ + d x d U โ d t d x โ = 0
m a + d U d x v = 0 ma + \frac{dU}{dx}v = 0 ma + d x d U โ v = 0
F = โ d U d x F = -\frac{dU}{dx} F = โ d x d U โ F โ = โ โ U \vec{F} = -\nabla U F = โ โ U
Equilibrium Points At equilibrium, F = 0 F = 0 F = 0 d U d x = 0 \frac{dU}{dx} = 0 d x d U โ = 0
Stable equilibrium: d 2 U d x 2 > 0 \frac{d^2U}{dx^2} > 0 d x 2 d 2 U โ > 0 U U U
Unstable equilibrium: d 2 U d x 2 < 0 \frac{d^2U}{dx^2} < 0 d x 2 d 2 U โ < 0 U U U
Neutral equilibrium: d 2 U d x 2 = 0 \frac{d^2U}{dx^2} = 0 d x 2 d 2 U โ = 0 U U U
Example: Potential Energy Curve U ( x ) = 1 2 k x 2 โ 1 6 b x 3 U(x) = \frac{1}{2}kx^2 - \frac{1}{6}bx^3 U ( x ) = 2 1 โ k x 2 โ 6 1 โ b x 3
Equilibrium points:
d U d x = k x โ 1 2 b x 2 = 0 \frac{dU}{dx} = kx - \frac{1}{2}bx^2 = 0 d x d U โ = k x โ 2 1 โ b x 2 = 0
x = 0 ย orย x = 2 k b x = 0 \text{ or } x = \frac{2k}{b} x = 0 ย orย x = b 2 k โ
Stability:
d 2 U d x 2 = k โ b x \frac{d^2U}{dx^2} = k - bx d x 2 d 2 U โ = k โ b x
At x = 0 x = 0 x = 0 d 2 U d x 2 = k > 0 \frac{d^2U}{dx^2} = k > 0 d x 2 d 2 U โ = k > 0
At x = 2 k b x = \frac{2k}{b} x = b 2 k โ d 2 U d x 2 = k โ 2 k = โ k < 0 \frac{d^2U}{dx^2} = k - 2k = -k < 0 d x 2 d 2 U โ = k โ 2 k = โ k < 0
Energy Diagrams Plot U ( x ) U(x) U ( x ) x x x E E E
Particle confined to regions where E โฅ U ( x ) E \geq U(x) E โฅ U ( x )
Turning points where E = U ( x ) E = U(x) E = U ( x )
Kinetic energy: K E = E โ U ( x ) KE = E - U(x) K E = E โ U ( x )
Non-Conservative Forces When non-conservative forces (friction, drag) are present:
W n c = ฮ K E + ฮ U = ฮ E W_{nc} = \Delta KE + \Delta U = \Delta E W n c โ = ฮ K E + ฮ U = ฮ E
Mechanical energy is not conserved; it decreases by work done against non-conservative forces.
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