3 Most FAQ’s of Work, Energy and Power Chapter in Inter 1st Year Physics (TS/AP)

8 Marks

LAQ-1 : State and prove law of conservation of energy in case of a freely falling body.

Introduction

When a body falls freely, only the force of gravity influences its motion, and it possesses both potential energy and kinetic energy. The law of conservation of energy states that the total mechanical energy of a system remains constant as long as no external forces are acting on it. Let’s understand how the conservation of mechanical energy holds for a freely falling body step by step.

1. Understanding Mechanical Energy: Mechanical energy (E) is the sum of potential energy (U) and kinetic energy (K) of the body. Therefore, E = K + U.
2. Expressing Potential Energy: Potential energy (U) is the negative of the work done by the force affecting the body, which in this case is gravity. For a body of mass (m) at height (h), potential energy is given by U = mgh, where g is the acceleration due to gravity.
3. Expressing Kinetic Energy: Kinetic energy (K) is the energy possessed by a body in motion. For a body of mass (m) moving with speed (v), its kinetic energy is given by K = 0.5 × m × v².
4. Dropping the Body: Consider dropping a ball of mass (m) from a height (H). Initially, the ball has potential energy U1 = m × g × H as its speed is zero.
5. At a Lower Height: Let the ball fall freely for some time (t) and reach a lower height (h). At this point, its potential energy becomes U2 = m × g × h.
6. Calculating Change in Potential Energy: The change in potential energy (ΔU) is equal to U2 − U1 = m × g × h − m × g × H = −m × g × (H−h).
7. Calculating Change in Kinetic Energy: We need to calculate the kinetic energy (K2) of the ball at height (h). To do this, we use the work-energy theorem, which states that the work done on a body is equal to the change in its kinetic energy. The work done (W) by gravity in this case is mg × x, where x is the displacement of the ball. Integrating the work done, we get 0.5 × m × v² = mg × (H−h), which simplifies to v² = 2g × (H−h).
8. Comparing Change in Potential and Kinetic Energy: We find that ΔK = −ΔU, which means the change in kinetic energy is equal in magnitude but opposite in sign to the change in potential energy.

Summary

Since ΔK = −ΔU, the total mechanical energy ΔE = ΔK + ΔU = 0. This shows that the mechanical energy of the freely falling body is conserved. In other words, as the body falls, the sum of its potential and kinetic energy remains constant, proving the law of conservation of mechanical energy.

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LAQ-2 : What are collisions? Explain the possible types of collisions? Develop the theory of one dimensional elastic collision.

Introduction

Collisions refer to events where two or more objects interact, resulting in a change in their motion. During a collision, there is a transfer of momentum and kinetic energy between the objects involved. Collisions can occur in various scenarios, such as in sports, car accidents, or at the microscopic level between particles.

Types of Collisions There are two main types of collisions based on the conservation laws they follow:

1. Elastic Collision:
   • In an elastic collision, both momentum and kinetic energy are conserved.
   • The total kinetic energy before the collision is equal to the total kinetic energy after the collision.
   • The objects involved in an elastic collision bounce off each other without any loss of energy.
   • Real-life examples include colliding billiard balls or bouncing balls.
2. Inelastic Collision:
   • In an inelastic collision, momentum is conserved, but kinetic energy is not conserved.
   • The objects involved may stick together or deform upon collision, resulting in a loss of kinetic energy.
   • Real-life examples include car crashes or objects sticking to a Velcro surface.

One-Dimensional Elastic Collision

Consider two objects of masses m₁​ and m₂​ moving along a straight line. Their initial velocities are v₁​ and v₂​ respectively, and their final velocities after the collision are v_1f​ and v_2f​ respectively.

1. The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision, expressed as: $$m_1v_1 + m_2v_2 = m_1v_{1f} + m_2v_{2f}​$$
2. The law of conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision, expressed as: $$\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$$

Solving for Final Velocities

To find the final velocities v_1f​ and v_2f​, we solve these two equations simultaneously. The solutions for the final velocities are:

$$v_{1f} = \frac{m_1 – m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2$$

$$v_{2f} = \frac{2m_1}{m_1 + m_2}v_1 + \frac{m_2 – m_1}{m_1 + m_2}v_2$$

Summary

Collisions involve interactions between objects that result in changes in their momentum and kinetic energy. The two main types are elastic and inelastic collisions. In one-dimensional elastic collisions, both momentum and kinetic energy are conserved, and the final velocities of the objects can be determined using equations derived from the conservation laws.

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LAQ-3 : Develop the notions of work and kinetic energy and show that  it leads to Work-Energy Theorem.

Introduction

The concepts of work and kinetic energy are fundamental in physics, especially in understanding the dynamics of objects under the influence of forces. Work is defined as the transfer of energy to an object via a force causing the object to move. Kinetic energy, on the other hand, is the energy that an object possesses due to its motion.

Defining Work

1. Work (W) is defined as the product of the force (F) applied to an object and the displacement (d) of the object in the direction of the force.
2. Mathematically, work is expressed as:
   • W = F ⋅ d ⋅ cos(θ)
   • Here, θ is the angle between the force vector and the direction of displacement.

Defining Kinetic Energy

1. Kinetic Energy (KE) is the energy of an object due to its motion.
2. It is given by the formula:
   • $$KE = \frac{1}{2}mv^2$$
   • Where m is the mass of the object, and v is its velocity.

Work-Energy Theorem

The Work-Energy Theorem is a crucial concept that connects the notions of work and kinetic energy. It states that the work done by all forces acting on an object is equal to the change in its kinetic energy.

1. Mathematically, the theorem is expressed as:
   • W = ΔKE
   • Where ΔKE is the change in kinetic energy of the object.
2. This theorem implies that when work is done on an object, it results in a change in the object’s kinetic energy. This can be an increase or decrease in kinetic energy, depending on the direction of the force relative to the object’s motion.

Application of Work-Energy Theorem

1. The Work-Energy Theorem helps in calculating the kinetic energy of an object when it is subjected to a certain amount of work by forces.
2. It also aids in understanding the energy transfer processes in various physical scenarios, such as in the case of a falling object, where gravitational force does work, converting potential energy into kinetic energy.

Summary

In summary, the development of the notions of work and kinetic energy leads directly to the Work-Energy Theorem. This theorem plays a crucial role in understanding how forces acting on an object lead to changes in its motion and energy states. The theorem essentially provides a link between the force applied to an object, the work done by this force, and the resultant change in the object’s kinetic energy.