Notes : Concept of Potential Energy and Conservative Forces , Force-Potential Energy Relation class 11 physics chapter 5 work , energy and power
1. Potential Energy
Definition
Potential Energy (PE) is the energy possessed by a body due to its position or configuration. It is often called stored energy because it can be converted into other forms of energy, especially kinetic energy.
Examples
- Water stored in a dam
- A stretched rubber band
- A compressed spring
- A stone held at a height above the ground
Spring Analogy
When a spring is compressed, work is done on it. This work gets stored inside the spring as potential energy. When the spring is released, the stored energy converts into kinetic energy.
2. Gravitational Potential Energy
Consider a body of mass \(m\) raised vertically through a height \(h\) near the Earth's surface.
Assumptions
- Height \(h\) is very small compared to Earth's radius.
- Acceleration due to gravity \(g\) remains constant.
\[ h \ll R_E \]
Step-by-Step Derivation
Step 1: Force Required to Lift the Body
\[ F = mg \]
where:
- \(m\) = mass of body
- \(g\) = acceleration due to gravity
Step 2: Calculate Work Done
\[ W = F \times h \]
Substituting \(F = mg\),
\[ W = (mg)h \]
\[ W = mgh \]
This work is stored as gravitational potential energy.
\[ V = mgh \]
\[ \boxed{V = mgh} \]
3. Conversion of Potential Energy into Kinetic Energy
Suppose the body is released from height \(h\).
Initial Conditions
\[ u = 0 \]
\[ K_i = 0 \]
Step 1: Apply Equation of Motion
\[ v^2 = u^2 + 2gh \]
Substituting \(u = 0\),
\[ v^2 = 0 + 2gh \]
\[ v^2 = 2gh \]
Step 2: Multiply Both Sides by \(\frac{1}{2}m\)
\[ \frac{1}{2}mv^2 = \frac{1}{2}m(2gh) \]
Step 3: Simplify
\[ \frac{1}{2}mv^2 = \frac{2mgh}{2} \]
\[ \frac{1}{2}mv^2 = mgh \]
Since
\[ V = mgh \]
Therefore,
\[ \boxed{\frac{1}{2}mv^2 = mgh} \]
Thus, gravitational potential energy converts completely into kinetic energy.
4. Relationship Between Force and Potential Energy
Step 1: Elementary Work Done
\[ dW = F\,dx \]
Step 2: Relation Between Work and Potential Energy
\[ dW = -dV \]
Step 3: Equate the Two Expressions
\[ Fdx = -dV \]
Step 4: Divide Both Sides by \(dx\)
\[ \frac{Fdx}{dx} = \frac{-dV}{dx} \]
Step 5: Simplify
\[ F = -\frac{dV}{dx} \]
\[ \boxed{F = -\frac{dV}{dx}} \]
Meaning of Negative Sign
The negative sign indicates that force always acts in the direction of decreasing potential energy.
5. Derivation of Work Done in Terms of Potential Energy
Starting from
\[ F = -\frac{dV}{dx} \]
Step 1: Multiply by \(dx\)
\[ Fdx = -\frac{dV}{dx}dx \]
Step 2: Cancel \(dx\)
\[ Fdx = -dV \]
Step 3: Replace \(Fdx\) with \(dW\)
\[ dW = -dV \]
Step 4: Integrate Both Sides
\[ \int_{x_i}^{x_f} dW = -\int_{x_i}^{x_f} dV \]
Step 5: Integrate
\[ W = -\left[V\right]_{x_i}^{x_f} \]
Step 6: Apply Limits
\[ W = -(V_f-V_i) \]
Step 7: Open the Bracket
\[ W = -V_f+V_i \]
\[ W = V_i-V_f \]
Step 8: Use Change in Potential Energy
\[ \Delta V = V_f-V_i \]
Therefore,
\[ W = -\Delta V \]
\[ \boxed{W = -\Delta V} \]
6. Verification for Gravitational Force
\[ V = mgh \]
Using
\[ F = -\frac{dV}{dh} \]
Substitute \(V = mgh\),
\[ F = -\frac{d(mgh)}{dh} \]
\[ F = -mg\frac{d(h)}{dh} \]
\[ \frac{dh}{dh}=1 \]
\[ F=-mg(1) \]
\[ F=-mg \]
\[ \boxed{F=-mg} \]
7. Conservative Forces
Definition
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.
Examples
- Gravitational force
- Electrostatic force
- Spring force
Inclined Plane Example
At the top:
\[ PE=mgh \]
At the bottom:
\[ KE=\frac12 mv^2 \]
Using conservation of energy,
\[ mgh=\frac12 mv^2 \]
\[ 2mgh=mv^2 \]
\[ 2gh=v^2 \]
\[ v=\sqrt{2gh} \]
The final velocity depends only on height \(h\), not on the path.
8. Non-Conservative Forces
Definition
A force is called non-conservative if the work done by it depends on the path followed.
Examples
- Friction
- Air resistance
- Viscous force
9. Dimensions and Units
Dimensional Formula
\[ \boxed{[ML^2T^{-2}]} \]
SI Unit
\[ \boxed{\text{Joule (J)}} \]
10. Frequently Asked Questions (FAQ)
Q1. What is potential energy?
Potential energy is the energy possessed by a body due to its position or configuration.
Q2. What is the SI unit of potential energy?
Joule (J).
Q3. What is the dimensional formula of potential energy?
\([ML^2T^{-2}]\)
Q4. What is gravitational potential energy?
\(V=mgh\)
Q5. What is a conservative force?
A force whose work done depends only on the initial and final positions.
Q6. What is a non-conservative force?
A force whose work done depends on the path followed.
11. MCQ Quiz
-
Potential energy is energy due to:
A) Velocity
B) Momentum
C) Position or configuration
D) Acceleration
Answer: C) Position or configuration -
The formula of gravitational potential energy is:
A) mg/h
B) mgh
C) ½mv²
D) gh
Answer: B) mgh -
The SI unit of potential energy is:
A) Newton
B) Watt
C) Joule
D) Pascal
Answer: C) Joule -
The relation between force and potential energy is:
A) \(F=\frac{dV}{dx}\)
B) \(F=-\frac{dV}{dx}\)
C) \(F=Vx\)
D) \(F=\frac{V}{x}\)
Answer: B) \(F=-\frac{dV}{dx}\) -
Which of the following is a conservative force?
A) Friction
B) Air resistance
C) Gravitational force
D) Viscous force
Answer: C) Gravitational force -
Which of the following is a non-conservative force?
A) Gravity
B) Electrostatic force
C) Spring force
D) Friction
Answer: D) Friction -
The work done by a conservative force equals:
A) \(\Delta V\)
B) \(-\Delta V\)
C) \(V_f+V_i\)
D) Zero
Answer: B) \(-\Delta V\) -
During free fall, potential energy converts into:
A) Heat Energy
B) Sound Energy
C) Kinetic Energy
D) Electrical Energy
Answer: C) Kinetic Energy -
The speed of a freely falling body is:
A) \(\sqrt{gh}\)
B) \(\sqrt{2gh}\)
C) \(gh\)
D) \(\sqrt{3gh}\)
Answer: B) \(\sqrt{2gh}\)
Quick Revision
- Potential energy is stored energy.
- Gravitational potential energy is \(mgh\).
- \(F=-\frac{dV}{dx}\)
- \(W=-\Delta V\)
- Gravity is a conservative force.
- Friction is a non-conservative force.
- SI Unit = Joule (J).
- Dimensional Formula = \([ML^2T^{-2}]\).
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