Chapter 7: Energy of a System – Physics for Scientists and Engineers

Energy of a System

Introduction to Energy

The concept of energy is central to physics and engineering, as every physical process in the universe involves energy. While many problems can be solved using Newton's Laws and associated principles, energy methods often provide simpler and more general solutions, especially for complex systems.

• Energy: A scalar quantity representing the ability to do work or produce change.

• Importance: Energy is involved in all physical processes, from mechanical motion to thermal interactions.

• Application: Energy concepts can simplify problem-solving compared to force-based approaches.

System Analysis Model

Physics often shifts from analyzing individual particles to considering entire systems. In this chapter, systems are introduced along with three primary ways to store energy.

• System: A collection of objects or particles considered together for analysis.

• System Boundary: The surface that separates the system from its environment.

• Energy Storage: Systems can store energy in different forms, such as kinetic, potential, and internal energy.

System Example

Consider an object in empty space subjected to a force. The object is the system, and the surface of the object is the system boundary. Forces from the environment act across this boundary.

• External Force: An influence from outside the system that can change the system's energy.

• System Boundary: Defines what is included in the system and what is considered the environment.

Work Done on a System

Work is a measure of energy transfer to or from a system by means of a force acting over a displacement.

• Definition: The work $W$ done by a constant force $F$ over a displacement $riangle r$ is given by:

$W = F riangle r \, \cos \theta$

• $\theta$: The angle between the force and the displacement vector.

• Work by Environment: Work is done by the environment on the system when a force acts across the system boundary.

Work Example

When an object undergoes a displacement $riangle r$ under a constant force $F$, the work done is:

$W = F riangle r \, \cos \theta$

• Only forces with a component along the displacement do work.

• Normal and gravitational forces may do zero work if perpendicular to displacement.

About Work

Work is a scalar quantity and can be positive or negative depending on the direction of force relative to displacement.

• Unit of Work: Joule (J)

• 1 Joule: $1 \text{ J} = 1 \text{ N} \cdot \text{m} = 1 \text{ kg} \cdot \text{m}^2 / \text{s}^2$

• Negative Work: Occurs when force opposes displacement (e.g., friction slowing a moving object).

Scalar Product of Two Vectors (Dot Product)

The scalar (dot) product of two vectors is fundamental in calculating work.

• Definition: $\vec{A} \cdot \vec{B} = AB \cos \theta$

• Application to Work: $W = \vec{F} \cdot \triangle \vec{r} = F \triangle r \cos \theta$

Example Problems

Example A

Calculate the work required to raise 15 kg of water from a well 12 m deep, assuming a constant upward acceleration of 0.7 m/s2.

• Application: Use the work formula and account for gravitational and acceleration forces.

Example B

A skier with mass 40 kg moves 20 m up a slope inclined at 15° to the horizontal. The tension in the rope pulling the skier is 250 N, and the rope makes a 30° angle with the slope. With a coefficient of kinetic friction $\mu_k = 0.1$, determine the work done by each force and the total work on the skier.

• Application: Resolve forces, calculate work for each, and sum for total work.

Work Done by a Varying Force

When the force is not constant, the total work is found by integrating the force over the displacement.

• Approximation: For small intervals where force is nearly constant, $W \approx F \triangle x$

• General Case: $W = \int_{x_1}^{x_2} F(x) \, dx$

• Graphical Interpretation: Work equals the area under the force vs. displacement curve.

Hooke's Law and Springs

Springs store energy through elastic deformation, described by Hooke's Law.

• Hooke's Law: $F_s = -kx$

• $k$: Spring constant (N/m), measures stiffness.

• $x$: Displacement from equilibrium position.

• Restoring Force: Always directed opposite to displacement.

Elastic Potential Energy

Energy stored in a spring is called elastic potential energy.

• Formula: $U = \frac{1}{2} k x^2$

• Properties: Zero when spring is relaxed ($x = 0$), maximum at maximum extension/compression, always positive.

Kinetic Energy

Kinetic energy is the energy of motion possessed by a particle or system.

• Formula: $K = \frac{1}{2} m v^2$

• Unit: Joule (J)

• Change in Kinetic Energy: Work done on a system can change its kinetic energy.

Work-Kinetic Energy Theorem

The net work done by all forces on a particle equals the change in its kinetic energy.

• Theorem: $\sum W = \triangle K$

• Application: Useful for analyzing motion under constant or varying forces.

Gravitational Potential Energy

Gravitational potential energy is associated with an object's position in a gravitational field.

• Formula: $U = mgh$

• Change in Potential Energy: $\triangle U = U_f - U_i = -W$

• Reference Point: The choice of zero potential energy is arbitrary, but differences are physically meaningful.

Elastic Potential Energy (Spring Systems)

Elastic potential energy is stored in a deformed spring and is given by:

• Formula: $U = \frac{1}{2} k x^2$

• Energy Storage: Only when the spring is stretched or compressed.

Energy Bar Charts

Energy bar charts are graphical tools to represent the distribution of energy types within a system.

• Vertical Axis: Amount of energy of each type.

• Horizontal Axis: Types of energy (kinetic, potential, elastic, etc.).

• Application: Useful for visualizing energy transformations and conservation.

Internal Energy and Friction

Friction converts mechanical energy into internal energy, increasing the temperature of the system.

• Internal Energy: Energy associated with microscopic motion and interactions within a system.

• Friction: Work done by friction is transformed into internal energy, often as heat.

• Energy Conservation: Total energy remains constant, but mechanical energy may decrease as internal energy increases.

Power

Power is the rate at which work is done or energy is transferred.

• Average Power: $P_{avg} = \frac{\triangle W}{\triangle t}$

• Instantaneous Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$

• Unit: Watt (W), where $1 \text{ W} = 1 \text{ J/s}$

Example Problems (Power)

• Elevator Example: Calculate the power required to raise a 2000 kg elevator attached to an 1800 kg counterweight at 0.4 m/s.

• Box Example: A person pushes a 40 kg box on a horizontal surface with $\mu_k = 0.2$ at constant speed over a distance of 6.5 m in 9 s. Find the power produced by the person.

Additional info: Some details and equations have been expanded for clarity and completeness, including explicit formulas and definitions for energy, work, and power.