Chapter 5: Work, Energy, and Power - Comprehensive NEET Physics Notes

1. Scalar Product of Two Vectors

1.1 Scalar Product Formula

The scalar product (dot product) of two vectors $\mathbf{A}$ and $\mathbf{B}$ is given by: $\mathbf{A}\cdot \mathbf{B}=AB\cos \theta$ where:

• $A$ and $B$ are the magnitudes of vectors $\mathbf{A}$ and $\mathbf{B}$ respectively.
• $\theta$ is the angle between the two vectors.

Explanation: The scalar product results in a scalar (a quantity with magnitude but no direction). It represents the product of the magnitude of one vector and the component of the other vector along the direction of the first vector.

Common Pitfalls: A common mistake is to forget that the scalar product is a scalar quantity, meaning it has no direction, unlike the vector product.

2. Work, Energy, and the Work-Energy Theorem

2.1 Work Done by a Force

The work $W$ done by a constant force $\mathbf{F}$ when it causes a displacement $\mathbf{d}$ is: $W=\mathbf{F}\cdot \mathbf{d}=Fd\cos \theta$ where:

• $\mathbf{F}$ is the force applied.
• $\mathbf{d}$ is the displacement caused by the force.
• $\theta$ is the angle between the force and the displacement.

Explanation: Work is a measure of energy transfer. When a force causes a displacement, the work done is the product of the force component in the direction of the displacement and the magnitude of the displacement.

Common Mistakes: Ensure the angle $\theta$ is correctly identified, especially when force and displacement are not aligned.

2.2 Kinetic Energy and Work-Energy Theorem

Kinetic energy $K$ of an object of mass $m$ moving with velocity $v$ is given by: $K=\frac{1}{2}m{v}^2$

The work-energy theorem states: $W_{\text{net}}=\Delta K=K_f-K_i$ where:

• $W_{\text{net}}$ is the net work done on the object.
• $K_f$ and $K_i$ are the final and initial kinetic energies, respectively.

Explanation: This theorem indicates that the work done by the net force on an object results in a change in its kinetic energy.

NEET Tip: In problems involving work and kinetic energy, always check for conservation of mechanical energy if only conservative forces are acting.

2.3 Potential Energy and Conservation of Mechanical Energy

Potential energy $V(x)$ for a conservative force is given by: $V(x)=-\int F(x)dx$

For gravitational potential energy near Earth's surface: $V(h)=mgh$ where:

• $m$ is the mass of the object.
• $g$ is the acceleration due to gravity.
• $h$ is the height above the reference point.

Explanation: Potential energy represents stored energy due to an object's position. The conservation of mechanical energy principle states that in the absence of non-conservative forces, the total mechanical energy (kinetic + potential) remains constant.

Real-life Application: The concept of gravitational potential energy is crucial in understanding phenomena like the working of hydroelectric power plants, where water at height converts potential energy into kinetic energy and then into electrical energy.

3. Power

Power $p$ is the rate at which work is done or energy is transferred: $P=\frac{dW}{dt}=\mathbf{F}\cdot \mathbf{v}$ where:

• $p$ is power.
• $W$ is work done.
• $t$ is time.
• $\mathbf{v}$ is velocity.

Explanation: Power measures how quickly work is done. It is a crucial concept in analyzing systems where energy transfer speed is important, such as in electrical appliances.

NEET Problem-Solving Strategy: In questions involving power, always identify the correct formula based on whether you are dealing with instantaneous power (using velocity) or average power (using total work and total time).

Quick Recap:

• Work: $W=\mathbf{F}\cdot \mathbf{d}=Fd\cos \theta$
• Kinetic Energy: $K=\frac{1}{2}m{v}^2$
• Work-Energy Theorem: $W_{\text{net}}=\Delta K$
• Potential Energy: $V(h)=mgh$
• Power: $P=\mathbf{F}\cdot \mathbf{v}$

Concept Connection:

• Physics to Chemistry: The concept of energy and its conservation is vital in thermodynamics, where it links to the study of enthalpy and internal energy.
• Physics to Biology: The principles of work and energy are also applicable in understanding biomechanics and energy transfer within living organisms.

Practice Questions:

1. Work Calculation: A force of 50 N is applied at an angle of 30° to the horizontal to move a block 5 m along the floor. Calculate the work done by the force.
   Solution:
2. • $W=Fd\cos \theta$
   • $W=50\times 5\times \cos 30^{\circ }$
   • $W=50\times 5\times \frac{\sqrt{3}}{2}$
   • $W=125\sqrt{3}\text{J}$
2. Kinetic Energy: A car of mass 1000 kg is moving at a speed of 20 m/s. Calculate its kinetic energy.
   Solution:
3. • $K=\frac{1}{2}m{v}^2$
   • $K=\frac{1}{2}\times 1000\times (20{)}^2$
   • $K=200,000\text{J}$
3. Power Calculation: A machine does 5000 J of work in 10 seconds. What is its power output?
   Solution:
4. • $P=\frac{W}{t}$
   • $P=\frac{5000}{10}$
   • $P=500\text{W}$

This summary captures the essential formulae and concepts from the chapter on Work, Energy, and Power, providing a solid foundation for NEET preparation.