⚡ Have You Ever Wondered?
Why does a roller-coaster car paused at the apex of a colossal steel drop possess the exact same total energy as when it barrels screaming through the...
Why does a roller-coaster car paused at the apex of a colossal steel drop possess the exact same total energy as when it barrels screaming through the bottom loop at 120 km/h? The Work-Energy Theorem and Conservation of Energy govern mechanical transformations.
यह अध्याय क्यों महत्वपूर्ण है
In Class 11 Physics, "Work, Energy and Power" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
अध्ययन से पूर्व (आवश्यक ज्ञान)
- Work, kinetic energy, and potential energy from Class 9.
- Dot product of vectors.
- Power in watts.
इस अध्याय के लक्ष्य
- Define Work done by a constant and variable force: $W = \vec{F} \cdot \vec{d} = \int F\, dx$.
- State and prove the Work-Energy Theorem: $W_{\text{net}} = \Delta K = K_f - K_i$.
- Distinguish between Conservative and Non-Conservative forces.
- Derive Potential Energy of a stretched Spring ($U = \frac{1}{2}kx^2$) and Gravitational Potential Energy ($mgh$).
- Analyze collisions in one dimension: Perfectly Elastic ($e = 1$) and Inelastic collisions ($e < 1$).
अध्याय रूपरेखा एवं प्रगति
1
1. Work Done & The Work-Energy Theo...
2
2. Conservative Forces & Potential...
3
3. Power & Collisions
सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन
1. Work Done & The Work-Energy Theorem
Work is the scalar product of force and displacement: $$\mathbf{W = \vec{F} \cdot \vec{d} = F d \cos\theta} \quad (\text{Variable force: } \mathbf{W = \int_{x_i}^{x_f} F(x)\, dx})$$ Work-Energy Theorem: The total work done by all forces acting on a particle equals the change in its kinetic energy: $$\mathbf{W_{\text{net}} = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2}$$
2. Conservative Forces & Potential Energy
A force is Conservative if the work done in a closed loop is strictly zero ($\oint \vec{F} \cdot d\vec{r} = 0$), meaning work depends solely on initial and final positions (Gravity, Electrostatics, Spring force).
• Potential Energy: $F = -\frac{dU}{dx}$.
• Spring Potential Energy: $\mathbf{U = \frac{1}{2}kx^2}$ ($k$ is spring constant).
3. Power & Collisions
- Power: Rate of doing work: $\mathbf{P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}}$ ($1\text{ Watt} = 1\text{ J/s}$; $1\text{ HP} = 746\text{ W}$).
- Elastic Collision: Both Linear Momentum AND Kinetic Energy are conserved ($e = 1$). For equal masses ($m_1 = m_2$), velocities interchange completely!
- Inelastic Collision: Momentum conserved, but Kinetic energy is lost to heat/sound ($e < 1$).
चित्रात्मक व्याख्या एवं मॉडल
Work, Energy and Power Master Matrix
Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture
1. Work Done & The Work-Energy Theorem • 2. Conservative Forces & Potential Energy
अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष
मुख्य बिंदु 1
Work-Energy Theorem: Equivalence connecting net mechanical work directly to kinetic delta.
मुख्य बिंदु 2
Conservative Force: Path-independent force derived from scalar potential gradients.
मुख्य बिंदु 3
Spring Potential Energy: Quadratic elastic energy storage $U = \frac{1}{2}kx^2$.
मुख्य बिंदु 4
Elastic Collision: Complete kinetic energy conservation with velocity interchange for equal masses.
मुख्य बिंदु 5
Coefficient of Restitution ($e$): Relative velocity of separation over relative velocity of approach.
स्व-मूल्यांकन अभ्यास (Check Your Understanding)
मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।
1
Prove the Work-Energy Theorem for a variable force.
उत्तर एवं व्याख्या देखें
उत्तर: $W = \int F\, dx = \int m\frac{dv}{dt}\, dx = \int m\, v\, dv = m \left[\frac{v^2}{2}\right]_{v_i}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = \Delta K$.
Integrated F dx = m v dv = ΔK.
2
A spring of force constant $k = 800\text{ N/m}$ has an extension of $5\text{ cm}$. What is the work done in increasing the extension from $5\text{ cm}$ to $15\text{ cm}$?
उत्तर एवं व्याख्या देखें
उत्तर: $W = \frac{1}{2}k(x_2^2 - x_1^2) = \frac{1}{2}(800)(0.15^2 - 0.05^2) = 400(0.0225 - 0.0025) = 400(0.02) = 8\text{ Joules}$.
8 Joules.
3
Show that in a head-on elastic collision between two identical masses, the bodies interchange their velocities after impact.
उत्तर एवं व्याख्या देखें
उत्तर: Conservation of momentum: $u_1 + u_2 = v_1 + v_2$. Conservation of KE / Restitution ($e=1$): $v_2 - v_1 = u_1 - u_2$. Adding: $2v_2 = 2u_1 \implies v_2 = u_1$; subtracting: $v_1 = u_2$. Velocities are interchanged.
v1 = u2 and v2 = u1; velocities swap.
4
A pump on the ground floor of a building can pump up water to fill a tank of volume $30\text{ m}^3$ in $15\text{ min}$. If the tank is $40\text{ m}$ above the ground and efficiency is 30%, how much electric power is consumed?
उत्तर एवं व्याख्या देखें
उत्तर: Mass $= 30 \times 1000 = 30,000\text{ kg}$. $W = mgh = 30000 \times 9.8 \times 40 = 11.76 \times 10^6\text{ J}$. Output Power $P_{\text{out}} = \frac{11.76 \times 10^6}{900\text{ s}} = 13.07\text{ kW}$. Electric power $= \frac{13.07}{0.30} \approx 43.56\text{ kW}$.
43.56 kW.
5
Under what conditions is the work done by a force zero?
उत्तर एवं व्याख्या देखें
उत्तर: Work $W = F d \cos\theta = 0$ when: (1) Force $F = 0$, (2) Displacement $d = 0$, or (3) Force is perpendicular to displacement ($\theta = 90^\circ$, such as centripetal force).
Zero force, zero displacement, or force perpendicular to motion.
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