Work-Energy Theorem and the Concept of Potential Energy

1. Work:

• Definition (NCERT Class 11, Chapter 6): Work is said to be done when a force causes a displacement of the object in its own direction.
• Distinction between positive and negative work: Positive work is done when the displacement is in the direction of the force, and negative work is done when the displacement is opposite to the direction of the force.
• Work done by a constant force: Work done by a constant force is given by $$W = F * s * cosθ$$ where,

• (F) is the magnitude of the force,
• (s) is the magnitude of the displacement of the point of application of the force, and
• (\theta) is the angle between the force vector and the displacement vector.

• Work done by a variable force: Work done by a variable force can be calculated using integration. The work done by a variable force (F(x)) over a displacement from (x_i) to (x_f) is given by $$W = \int_{x_i}^{x_f} F(x) dx$$

• Work done by non-conservative forces: Non-conservative forces, such as friction, do not conserve mechanical energy. The work done by non-conservative forces is dependent on the path taken by the object and cannot be expressed as a function of position only.

2. Energy:

• Different forms of energy:

• Kinetic energy: Energy possessed by an object due to its motion. It is given by $$K = \frac{1}{2} mv^2$$ where,

• (m) is the mass of the object, and

• (v) is its speed.

• Potential energy: Energy possessed by an object due to its position or configuration.

• Internal energy: Energy associated with the microscopic constituents of a system, such as the motion of atoms and molecules.

• Relationship between work done on a system and change in its energy: According to the work-energy theorem, the net work done on a system is equal to the change in its total energy. This can be expressed as: $$W_{net} = ΔE$$ where,

• (W_{net}) is the net work done on the system, and
• (ΔE) is the change in the total energy of the system.

3. Potential Energy:

• Gravitational potential energy (NCERT Class 11, Chapter 7): Energy possessed by an object due to its position in a gravitational field. It is given by $$U_g = mgh$$ where,

• (m) is the mass of the object,
• (g) is the acceleration due to gravity, and
• (h) is the vertical height of the object above a reference point.

• Elastic potential energy (NCERT Class 11, Chapter 9): Energy stored in a deformed elastic object, such as a stretched spring. It is given by $$U_e = \frac{1}{2}kx^2$$ where,

• (k) is the spring constant, and
• (x) is the amount of deformation.

• Relationship between conservative forces and potential energy: Conservative forces are forces for which the work done by the force is independent of the path taken by the object. The work done by conservative forces can be expressed as a function of position, and the negative of the work done by a conservative force is equal to the change in potential energy.

4. Work-Energy Theorem (NCERT Class 11, Chapter 6):

• Statement: The net work done on an object is equal to the change in its kinetic energy.
• Mathematical expression: $$W_{net} = ΔK$$ where,

• (W_{net})is the net work done on the object, and
• (ΔK) is the change in the kinetic energy of the object.

5. Conservation of Energy (NCERT Class 11, Chapter 7):

• Law of conservation of energy: The total energy of an isolated system remains constant, although it may be transformed from one form to another.
• Various forms of energy that can be interconverted: Mechanical energy, thermal energy, electrical energy, chemical energy, and more.
• Applications of the conservation of energy principle: Solving problems involving energy transformations and conservation of energy.

6. Potential Energy Diagrams:

• Graphical representations of potential energy as a function of position.
• Identifying equilibrium points, stable and unstable equilibrium, and understanding energy conservation principles visually.