Notes : Conservation of Mechanical Energy and Conservative Forces

Notes : Conservation of Mechanical Energy and Conservative Forces Class 11 physics chapter 5 work 

The law of conservation of mechanical energy is one of the most important principles in physics. It states that when only conservative forces act on a body, the sum of its kinetic energy and potential energy remains constant throughout the motion.

Mechanical Energy

Mechanical energy is the total energy possessed by a body due to its motion and position.

\[ E = K + V \]

where,

• \(E\) = Mechanical Energy
• \(K\) = Kinetic Energy
• \(V\) = Potential Energy

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Derivation of the Law of Conservation of Mechanical Energy

Step 1: Relation Between Conservative Force and Potential Energy

For a conservative force,

\[ F(x)=-\frac{dV}{dx} \]

Multiplying both sides by a small displacement \(dx\),

\[ F(x)\,dx=-\frac{dV}{dx}\,dx \]

Since,

\[ \frac{dV}{dx}\,dx=-dV \]

Therefore,

\[ F(x)\,dx=-dV \]

Step 2: Work Done by the Force

The infinitesimal work done by the force is

\[ dW=F(x)\,dx \]

Substituting \(F(x)\,dx=-dV\),

\[ dW=-dV \]

Step 3: Apply the Work-Energy Theorem

According to the Work-Energy Theorem,

\[ dW=dK \]

Substituting \(dW=-dV\),

\[ dK=-dV \]

Bringing all terms to one side,

\[ dK+dV=0 \]

Using the property of differentials,

\[ d(K+V)=0 \]

Step 4: Integration

Integrating both sides,

\[ \int d(K+V)=\int 0\,dx \]

The left-hand side becomes

\[ (K+V)\Big|_{i}^{f} \]

and the right-hand side becomes

\[ 0 \]

Therefore,

\[ (K_f+V_f)-(K_i+V_i)=0 \]

\[ K_f+V_f=K_i+V_i \]

Hence,

\[ K+V=\text{constant} \]

Thus, the total mechanical energy remains conserved.

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Conservative Force

A force is called conservative if the work done by it depends only on the initial and final positions and not on the path followed.

Conditions for a Conservative Force

1. Force Can Be Derived from Potential Energy

\[ F(x)=-\frac{dV}{dx} \]

2. Work Depends Only on Initial and Final Positions

\[ W=V(x_i)-V(x_f) \]

3. Work Done Over a Closed Path is Zero

\[ \oint \vec{F}\cdot d\vec{r}=0 \]

Examples of Conservative Forces

• Gravitational Force
• Electrostatic Force
• Spring Force

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Example: Freely Falling Ball

Consider a ball of mass \(m\) dropped from a cliff of height \(H\).

At the Top of the Cliff (Height = H)

Velocity:

\[ v=0 \]

Kinetic Energy:

\[ K=0 \]

Potential Energy:

\[ V=mgH \]

Total Mechanical Energy:

\[ E_H=mgH \]

At an Intermediate Height h

Velocity = \(v_h\)

Kinetic Energy:

\[ K=\frac{1}{2}mv_h^2 \]

Potential Energy:

\[ V=mgh \]

Total Energy:

\[ E_h=\frac{1}{2}mv_h^2+mgh \]

Just Before Reaching the Ground

Velocity = \(v_f\)

Potential Energy:

\[ V=0 \]

Kinetic Energy:

\[ K=\frac{1}{2}mv_f^2 \]

Total Energy:

\[ E_0=\frac{1}{2}mv_f^2 \]

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Finding Final Velocity Using Energy Conservation

Since energy is conserved,

\[ E_H=E_0 \]

\[ mgH=\frac{1}{2}mv_f^2 \]

Cancelling \(m\),

\[ gH=\frac{1}{2}v_f^2 \]

Multiplying by 2,

\[ 2gH=v_f^2 \]

Taking square root,

\[ v_f=\sqrt{2gH} \]

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Finding Velocity at Height h

Using conservation of energy,

\[ E_H=E_h \]

\[ mgH=\frac{1}{2}mv_h^2+mgh \]

Subtracting \(mgh\) from both sides,

\[ mgH-mgh=\frac{1}{2}mv_h^2 \]

Taking common factor \(mg\),

\[ mg(H-h)=\frac{1}{2}mv_h^2 \]

Cancelling \(m\),

\[ g(H-h)=\frac{1}{2}v_h^2 \]

Multiplying by 2,

\[ 2g(H-h)=v_h^2 \]

Taking square root,

\[ v_h=\sqrt{2g(H-h)} \]

Here, \((H-h)\) represents the distance fallen by the ball.

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Physical Significance of Conservation of Mechanical Energy

• Potential energy decreases during free fall.
• Kinetic energy increases during free fall.
• The loss in potential energy equals the gain in kinetic energy.
• Total mechanical energy remains constant.
• Energy conservation provides an easier method than kinematic equations for solving motion problems.

\[ -\Delta V=\Delta K \]

Therefore,

\[ K+V=\text{constant} \]

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Frequently Asked Questions (FAQs)

What is mechanical energy?

Mechanical energy is the sum of kinetic energy and potential energy.

What is the law of conservation of mechanical energy?

It states that total mechanical energy remains constant when only conservative forces act.

What is a conservative force?

A force whose work done depends only on initial and final positions.

Why is gravity a conservative force?

Because its work depends only on the starting and ending positions, not on the path.

What is the SI unit of mechanical energy?

Joule (J).

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Multiple Choice Questions (MCQs)

1. Mechanical energy is equal to:
   A) Force + Momentum
   B) Kinetic Energy + Potential Energy
   C) Work + Power
   D) Force × Distance
   Answer: B
2. Which of the following is a conservative force?
   A) Friction
   B) Air Resistance
   C) Gravitational Force
   D) Viscous Force
   Answer: C
3. Work done by a conservative force depends on:
   A) Time
   B) Velocity
   C) Path
   D) Initial and Final Positions
   Answer: D
4. The work done by a conservative force over a closed path is:
   A) Positive
   B) Negative
   C) Zero
   D) Infinite
   Answer: C
5. The final velocity of a body dropped from height H is:
   A) \(\sqrt{gH}\)
   B) \(\sqrt{2gH}\)
   C) \(2gH\)
   D) \(gH\)
   Answer: B