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Work, Energy and Power - Conservative and Non-conservative Forces

Grade 11ICSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Conservative Force is a force for which the work done in moving a particle between two points is independent of the path taken. Examples include gravitational force, electrostatic force, and the elastic spring force.

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The work done by a conservative force along any closed loop is zero: ∮F⃗⋅dr⃗=0\oint \vec{F} \cdot d\vec{r} = 0.

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Potential Energy (UU) is defined only for conservative forces. The change in potential energy is equal to the negative of the work done by the internal conservative force: ΔU=−Wc\Delta U = -W_{c}.

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A Non-conservative Force is a force where the work done depends on the path taken. Examples include friction, air resistance, and viscous drag.

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Work done by non-conservative forces is not recoverable as potential energy; it is usually dissipated as heat, sound, or light. For these forces, ∮F⃗⋅dr⃗≠0\oint \vec{F} \cdot d\vec{r} \neq 0.

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The Law of Conservation of Mechanical Energy states that if only conservative forces act on a system, the sum of kinetic energy (KK) and potential energy (UU) remains constant: E=K+U=constantE = K + U = \text{constant}.

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The Work-Energy Theorem in the presence of non-conservative forces states: Wtotal=Wc+Wnc=ΔKW_{total} = W_{c} + W_{nc} = \Delta K. This can be rewritten as Wnc=ΔK+ΔU=ΔEW_{nc} = \Delta K + \Delta U = \Delta E.

📐Formulae

W=∫r1r2F⃗⋅dr⃗W = \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r}

∮F⃗conservative⋅dr⃗=0\oint \vec{F}_{conservative} \cdot d\vec{r} = 0

Fx=−dUdxF_x = -\frac{dU}{dx}

ΔU=Uf−Ui=−∫xixfF(x)dx\Delta U = U_f - U_i = -\int_{x_i}^{x_f} F(x) dx

K1+U1=K2+U2+WfrictionK_1 + U_1 = K_2 + U_2 + W_{friction}

F⃗=−(∂U∂xi^+∂U∂yj^+∂U∂zk^)\vec{F} = -\left( \frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k} \right)

💡Examples

Problem 1:

Determine if the force field defined by F⃗=(2xy+z3)i^+x2j^+3xz2k^\vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k} is conservative. If a potential energy function exists, find it.

Solution:

A force is conservative if ∇⃗×F⃗=0\vec{\nabla} \times \vec{F} = 0. Checking components: ∂Fx∂y=2x\frac{\partial F_x}{\partial y} = 2x and ∂Fy∂x=2x\frac{\partial F_y}{\partial x} = 2x. Similarly, ∂Fx∂z=3z2\frac{\partial F_x}{\partial z} = 3z^2 and ∂Fz∂x=3z2\frac{\partial F_z}{\partial x} = 3z^2. Since partial derivatives match, the force is conservative. To find UU, we use Fx=−∂U∂xF_x = -\frac{\partial U}{\partial x}. Integrating −(2xy+z3)-(2xy + z^3) gives U=−x2y−xz3+CU = -x^2y - xz^3 + C.

Explanation:

This example demonstrates the mathematical test for a conservative force (curl being zero) and its relationship with the scalar potential field UU.

Problem 2:

A block of mass m=2 kgm = 2\text{ kg} is pushed 5 m5\text{ m} along a horizontal surface where the coefficient of kinetic friction is μk=0.3\mu_k = 0.3. It is then pushed back to its starting point. Calculate the total work done by friction.

Solution:

Friction is a non-conservative force. For the forward trip: Wf1=−fk⋅d=−(μkmg)d=−(0.3×2×9.8)×5=−29.4 JW_{f1} = -f_k \cdot d = -(\mu_k mg)d = -(0.3 \times 2 \times 9.8) \times 5 = -29.4\text{ J}. For the return trip, friction again opposes motion: Wf2=−29.4 JW_{f2} = -29.4\text{ J}. Total work Wtotal=Wf1+Wf2=−58.8 JW_{total} = W_{f1} + W_{f2} = -58.8\text{ J}.

Explanation:

Because friction is non-conservative, the work done over a closed path is not zero. The work depends on the total distance traveled (10 m10\text{ m}) rather than the displacement (0 m0\text{ m}).

Problem 3:

The potential energy of a system is given by U(x)=ax2−bxU(x) = \frac{a}{x^2} - \frac{b}{x}. Find the expression for the conservative force F(x)F(x) acting on the system.

Solution:

Using the relation F(x)=−dUdxF(x) = -\frac{dU}{dx}: F(x)=−ddx(ax−2−bx−1)F(x) = -\frac{d}{dx} \left( ax^{-2} - bx^{-1} \right) F(x)=−(−2ax−3+bx−2)F(x) = -\left( -2ax^{-3} + bx^{-2} \right) F(x)=2ax3−bx2F(x) = \frac{2a}{x^3} - \frac{b}{x^2}

Explanation:

The force is the negative gradient of the potential energy. This is a fundamental property of conservative systems.

Conservative and Non-conservative Forces Revision - Class 11 Physics ICSE