Work and Energy: Principles, Calculations, and Applications

Work and Energy

Introduction to Energy and Its Conservation

Energy is a fundamental concept in physics, representing a conserved quantity in all physical processes. The law of conservation of energy states that the total energy of a closed system remains constant, regardless of the transformations it undergoes. This principle is not a description of a mechanism but rather a statement about the invariance of a calculated quantity through all changes in nature.

• Kinetic Energy (T): Energy due to motion, given by $T = \frac{1}{2}mv^2$.

• Potential Energy (V): Energy due to interactions between objects, such as gravitational, elastic, or electric potential energy.

• Work (U): The process of energy transfer via force acting over a distance.

• Conservation Principle: The total energy (kinetic + potential) in a closed system is conserved.

Examples of Potential Energy:

• Gravitational: $V_g = mgy$

• Elastic (Spring): $V_s = \frac{1}{2}k\Delta x^2$

• Electric (Two Charges): $V_{elec} = k\frac{q_1q_2}{r}$, where $k = 9 \times 10^9 \ \mathrm{N\,m^2/C^2}$

Additional info: The Feynman block analogy illustrates conservation: the sum of all forms of energy remains constant, even as energy shifts between forms.

Work of a Force

Definition and Calculation of Work

Work is defined as the energy transferred to or from an object via the application of force along a displacement. For a force $\vec{F}$ acting on an object displaced by $d\vec{r}$, the infinitesimal work $dU$ is:

• $dU = \vec{F} \cdot d\vec{r} = F\,dr\cos\theta$

• For a finite displacement from point 1 to point 2: $U_{1 \to 2} = \int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r} = \int_{s_1}^{s_2} F\,ds\cos\theta$

Properties of Work:

• Work is a scalar quantity.

• SI unit: Joule (J), where $1\,\mathrm{J} = 1\,\mathrm{N}\cdot1\,\mathrm{m}$.

• The sign of work depends on the angle $\theta$ between force and displacement:

  • $0 \leq \theta < 90^\circ$: $\cos\theta > 0$ (work is positive)

  • $90^\circ < \theta \leq 180^\circ$: $\cos\theta < 0$ (work is negative)

  • $\theta = 90^\circ$: $\cos\theta = 0$ (work is zero)

• If force is constant: $U_{1 \to 2} = \vec{F} \cdot \vec{r}_{12}$

Work Done by Specific Forces

Work of Gravity (Weight)

The work done by gravity is positive when an object moves downward ($y_1 > y_2$), as the force and displacement are parallel. The work depends only on the initial and final heights, not the path taken:

• $U_{1 \to 2}^{(g)} = m g (y_1 - y_2)$

Work of an Elastic Force (Spring)

The work done by a spring force is positive when the spring returns to equilibrium ($x_1 > x_2$). The force is $\vec{F}_s(x) = -kx\hat{i}$, and the work is:

• $U_{1 \to 2}^{(s)} = \int_{x_1}^{x_2} (-kx) dx = \frac{1}{2}k x_1^2 - \frac{1}{2}k x_2^2$

Work-Energy Principle

Statement and Applications

The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy. This principle provides a powerful tool for solving problems involving forces and motion, especially when the goal is to find velocity rather than acceleration.

• Kinetic Energy: $T = \frac{1}{2}mv^2$

• Work-Energy Theorem: $\Delta T = T_2 - T_1 = U_{1 \to 2}$

• In words: The final kinetic energy equals the initial kinetic energy plus the work done by all forces as the object moves from its initial to final position.

• This approach is especially useful for finding speeds, but does not capture all information about normal forces (which are handled by Newton's second law in the normal direction).

Additional info: For a system of particles or when friction is involved, refer to the extended work-energy principle in your textbook (Section 14-3).

Summary Table: Types of Energy and Work

Type	Formula	Physical Meaning
Kinetic Energy	$T = \frac{1}{2}mv^2$	Energy due to motion
Gravitational Potential Energy	$V_g = mgy$	Energy due to position in a gravitational field
Elastic Potential Energy	$V_s = \frac{1}{2}k\Delta x^2$	Energy stored in a stretched or compressed spring
Electric Potential Energy	$V_{elec} = k\frac{q_1q_2}{r}$	Energy due to interaction between charges
Work (General)	$U = \int \vec{F} \cdot d\vec{r}$	Energy transferred by force over displacement