Work, Energy, Power, and Momentum: Structured Study Notes

Work and Energy

Definition of Work

Work is defined as the product of the force applied to an object and the displacement of the object in the direction of the force.

• Formula: $W = \vec{F} \cdot d\vec{x}$

• Dot Product: The work done by a constant force is the dot product of the force and displacement vectors.

• Scalar Product: $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$

• Example: If $\vec{A} = (A_x, A_y)$ and $\vec{B} = (B_x, B_y)$, then $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y$

Work Done by a Variable Force

When the force varies with position, work is calculated using integration.

• Formula: $W = \int_{x_i}^{x_f} F(x) dx$

• Graphical Interpretation: The area under the force vs. displacement curve represents the work done.

Work Done by a Spring

The work done by a spring force is determined by Hooke's Law.

• Hooke's Law: $F = -kx$

• Work by Spring: $W = \int_{x_i}^{x_f} (-kx) dx = \frac{1}{2}k(x_i^2 - x_f^2)$

• Example: Calculating work for a spring compressed or stretched from $x_i$ to $x_f$.

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion.

• Formula: $K = \frac{1}{2}mv^2$

Work-Energy Theorem

The net work done on an object is equal to the change in its kinetic energy.

• Formula: $W_{net} = \Delta K = K_f - K_i$

• Application: Used to solve problems involving forces and motion.

Power

Definition of Power

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

• Formula: $P = \frac{dW}{dt}$

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

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

Potential Energy

Definition and Types

Potential energy is the energy stored in an object due to its position or configuration.

• Gravitational Potential Energy: $U_g = mgh$

• Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$

Conservative and Non-Conservative Forces

Conservative forces allow the definition of potential energy, while non-conservative forces (like friction) dissipate energy.

• Conservative Force: Work done is path-independent.

• Non-Conservative Force: Work done depends on the path taken.

Work Done by Conservative Forces

• Formula: $W_{cons} = -\Delta U$

Conservation of Energy

Principle of Conservation of Mechanical Energy

The total mechanical energy (kinetic + potential) of a system remains constant if only conservative forces act.

• Formula: $K_i + U_i = K_f + U_f$

• Application: Used to solve problems involving energy transformations.

Linear Momentum and Collisions

Definition of Linear Momentum

Linear momentum is the product of an object's mass and velocity.

• Formula: $\vec{p} = m\vec{v}$

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• Formula: $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$

Conservation of Linear Momentum

In a closed system, the total linear momentum remains constant if no external forces act.

• Formula: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$

Collisions

Collisions are classified as elastic or inelastic based on whether kinetic energy is conserved.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

• Perfectly Inelastic Collision: Colliding objects stick together after collision.

Collisions in One Dimension

• Equations:

  • Momentum: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

  • Kinetic Energy (elastic): $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$

Collisions in Two Dimensions

• Momentum is conserved in both x and y directions.

Tables

Comparison of Conservative and Non-Conservative Forces

Type of Force	Work Path Dependence	Potential Energy Defined?
Conservative	No	Yes
Non-Conservative	Yes	No

Types of Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?
Elastic	Yes	Yes
Inelastic	Yes	No
Perfectly Inelastic	Yes	No

Examples and Applications

• Work by a Spring: Calculating work done when compressing or stretching a spring.

• Energy Conservation: Using $K_i + U_i = K_f + U_f$ to solve for unknown velocities or heights.

• Momentum in Collisions: Solving for final velocities after collision using conservation laws.

Additional info: These notes cover topics from Ch 06 (Work & Kinetic Energy), Ch 07 (Potential Energy & Conservation), and Ch 08 (Momentum, Impulse, and Collisions) of a college physics curriculum. The content is structured to provide a comprehensive overview suitable for exam preparation.