Table of Contents

1. 1. Definition of Energy
2. 2. Potential Energy & Case Analysis
3. 3. Elastic Potential Energy of a Spring
4. 4. Kinetic Energy & Derivation
5. 5. Work-Energy Theorem
6. 6. Conceptual Questions & Solved Numerical Examples
   • Part A: Conceptual Understanding Questions (Q1 – Q10)
   • Part B: Solved Numerical Examples (Problems 1 – 10)

1. Definition of Energy

Definition: Energy is defined as the total capacity of a physical system to perform work. It is a scalar quantity, meaning it possesses magnitude but no direction.

SI Unit: Joule ($\text{J}$), where $1\text{ J} = 1\text{ N}\cdot\text{m} = 1\text{ kg}\cdot\text{m}^2/\text{s}^2$

CGS Unit: Erg, where $1\text{ Joule} = 10^7\text{ Ergs}$

Dimensional Formula: $[\text{M}^1 \text{L}^2 \text{T}^{-2}]$

2. Potential Energy & Case Analysis

3. Elastic Potential Energy of a Spring

Figure 1: Spring-mass system at rest ($x=0$) and stretched by displacement $x$.

1. The small work done $dW$ by an external force to stretch the spring by an infinitesimal distance $dx$ against the restoring force is:

$$dW = F_{\text{ext}} \, dx = kx \, dx$$

2. Integrating from initial position $x = 0$ to final elongation $x$:

$$W = \int_{0}^{x} kx \, dx = k \left[ \frac{x^2}{2} \right]_{0}^{x} = \frac{1}{2} k x^2$$

3. Stored Elastic Potential Energy ($U$): $U = \frac{1}{2} k x^2$

4. Kinetic Energy & Derivation

1. Consider a body of mass $m$ initially at rest ($u = 0$). A constant force $F$ acts on it, accelerating it to velocity $v$ over displacement $s$.

2. Using Newton’s second law of motion: $F = m a$

3. From the third equation of motion ($v^2 = u^2 + 2as$):

$$v^2 = 0 + 2as \implies a s = \frac{v^2}{2}$$

4. Substituting force and displacement into the Work formula ($W = F s$):

$$W = (m a) s = m (a s) = m \left( \frac{v^2}{2} \right) = \frac{1}{2} m v^2$$

5. Since work done equals kinetic energy gained: $K = \frac{1}{2} m v^2$

5. Work-Energy Theorem

The net work done by all forces (conservative, non-conservative, internal, and external) acting on a body is equal to the net change in its kinetic energy.

$$W_{\text{net}} = \Delta K = K_f – K_i = \frac{1}{2}m v^2 – \frac{1}{2}m u^2$$

1. Instantaneous rate of change of kinetic energy:

$$\frac{dK}{dt} = \frac{d}{dt} \left(\frac{1}{2} m v^2\right) = m v \frac{dv}{dt} = m a v = F v = F \frac{dx}{dt}$$

2. Canceling $dt$ on both sides yields: $dK = F dx$

3. Integrating from initial state $(x_i, K_i)$ to final state $(x_f, K_f)$:

$$\int_{K_i}^{K_f} dK = \int_{x_i}^{x_f} F dx \implies K_f – K_i = W_{\text{net}}$$

6. Conceptual Questions & Solved Numerical Examples

Q1. Can a body have kinetic energy without having momentum?

No. Kinetic energy is related to momentum by $K = \frac{p^2}{2m}$. If momentum $p = 0$, velocity $v = 0$, so kinetic energy must be zero.

Q2. If the momentum of a body is doubled, by what factor does its kinetic energy increase?

Since $K \propto p^2$, doubling the momentum ($2p$) increases kinetic energy by $2^2 = \mathbf{4}$ times (quadrupled).

Q3. Can potential energy of a body be negative? Give an example.

Yes. Potential energy depends on the reference frame. For instance, gravitational potential energy below a chosen zero-level surface is negative ($U = -mgh$).

Q4. Does the Work-Energy Theorem hold true in non-inertial frames of reference?

Yes, provided the work done by pseudo forces is included in calculating $W_{\text{net}}$.

Q5. What happens to the potential energy of a spring when it is compressed vs. stretched?

In both cases, potential energy increases because $U = \frac{1}{2} k x^2 \ge 0$, regardless of whether $x$ is positive or negative.

Q6. A heavy body and a light body have equal kinetic energies. Which one has greater momentum?

Since $p = \sqrt{2mK}$, for equal kinetic energy $K$, momentum is proportional to $\sqrt{m}$. The heavier body has greater momentum.

Q7. Can kinetic energy ever be negative?

No. Mass $m > 0$ and $v^2 \ge 0$, so kinetic energy is strictly non-negative ($K \ge 0$).

Q8. When a bullet gets embedded in a wooden block, what happens to its kinetic energy?

Most kinetic energy is converted into heat, sound, and non-conservative work doing structural deformation.

Q9. How does spring stiffness constant ($k$) affect work required to stretch a spring?

A stiffer spring (larger $k$) requires proportionally more work ($W = \frac{1}{2} k x^2$) for the same displacement $x$.

Q10. Is kinetic energy dependent on the frame of reference?

Yes. Velocity is relative to the frame of reference, so kinetic energy value varies by observer motion.

Problem 1: Kinetic Energy Calculation

Question: Calculate the kinetic energy of an object of mass $10\text{ kg}$ moving with a uniform velocity of $6\text{ m/s}$.

Problem 2: Elastic Potential Energy of a Compressed Spring

Question: A spring with a spring constant $k = 500\text{ N/m}$ is compressed by $0.05\text{ m}$ from its natural length. Find the elastic potential energy stored in the spring.

Problem 3: Work-Energy Theorem for a Braking Vehicle

Question: A car of mass $1000\text{ kg}$ traveling at a speed of $20\text{ m/s}$ is brought to rest by applying brakes. Calculate the net work done on the car by the braking force.

Problem 4: Relation Between Momentum & Kinetic Energy

Question: A body of mass $4\text{ kg}$ has a linear momentum of $20\text{ kg}\cdot\text{m/s}$. Find its kinetic energy.

Problem 5: Gravitational Potential Energy

Question: Calculate the potential energy gained by a $5\text{ kg}$ object when raised vertically to a height of $12\text{ m}$. (Take $g = 9.8\text{ m/s}^2$)

Problem 6: Work Required to Change Speed

Question: How much work must be done to accelerate a $2\text{ kg}$ block from $5\text{ m/s}$ to $10\text{ m/s}$ on a frictionless horizontal plane?

Problem 7: Spring Extension Work Ratio

Question: It takes $10\text{ J}$ of work to stretch a spring by a displacement $x$. How much total energy is stored in the spring when it is stretched by a total displacement of $3x$ from equilibrium?

Problem 8: Work Done by Variable Force

Question: A variable force $F(x) = 6x\text{ N}$ acts on a $3\text{ kg}$ mass moving along the x-axis from $x = 0$ to $x = 2\text{ m}$. Find the total change in kinetic energy of the mass.

Problem 9: Velocity from Conservation of Energy

Question: A body of mass $m$ is dropped from rest from a height of $20\text{ m}$. Using energy conservation, calculate its velocity right before striking the ground. (Take $g = 10\text{ m/s}^2$)

Problem 10: Percentage Increase in Kinetic Energy

Question: If the velocity of a moving object is increased by $50\%$, determine the percentage increase in its kinetic energy.