Work, Energy, and Momentum: Review and Practice Problems

Work, Energy, and Momentum

Key Equations and Concepts

This section summarizes the fundamental equations and principles related to work, energy, and momentum in classical mechanics. These concepts are essential for understanding the behavior of objects under the influence of forces.

• Newton's Second Law: $\vec{F}_{\text{net}} = m\vec{a}$

• Friction Force: $F_f = \mu_k F_n$

• Work Done by a Constant Force: $W_{\text{ext}} = F_{\text{ext}} d \cos \theta$

• Work-Energy Theorem: $W_{\text{net}} = \Delta KE = KE_f - KE_i$

• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Potential Energy (PE): $PE_g = mgh$ $PE_{\text{elastic}} = \frac{1}{2}kx^2$

• Mechanical Energy (ME): $ME = KE + PE$

• Conservation of Mechanical Energy: $ME_i = ME_f$

• Power: $P = \frac{W}{\Delta t}$

• Impulse-Momentum Theorem: $\vec{J} = \int \vec{F}_{\text{net}} dt = \Delta \vec{p}$ $\vec{p} = m\vec{v}$

• Conservation of Linear Momentum: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Conceptual Understanding

Work and Energy Transformations

• Work by Gravity: The work done by gravity depends on the direction of motion relative to the gravitational force. It is positive when the object moves downward, negative when moving upward, and zero if the displacement is perpendicular to gravity.

• Pendulum Motion: As a pendulum swings, energy transforms between kinetic and potential forms, but the total mechanical energy remains constant (if air resistance is negligible).

• Stopping a Car: When a car stops, the work done by friction is negative, as it removes kinetic energy from the system.

• Kinetic Energy Examples: An object with nonzero kinetic energy must be in motion (e.g., a moving car, a falling rock).

Mechanical Energy in Systems

• Roller Coaster Example: The total mechanical energy (KE + PE) is greatest at the highest point if no energy is lost to friction. The speed is greatest where potential energy is lowest.

• Projectile Motion: As a projectile rises, kinetic energy decreases and potential energy increases. At the same height, the potential energy is the same, but kinetic energy may differ depending on the velocity.

• Elastic and Inelastic Collisions: In elastic collisions, both kinetic energy and momentum are conserved. In inelastic collisions, only momentum is conserved.

Problem-Solving Applications

Work and Energy Calculations

• Work by Multiple Forces: When multiple forces act at angles, resolve each force into components and use vector addition to find the total work done.

• Minimum Energy for Speed Change: The minimum energy required to change an object's speed is equal to the change in kinetic energy.

• Work on an Incline: For objects pulled at an angle, use trigonometry to resolve the force and calculate the work done along the direction of motion.

• Friction and Initial Speed: To find the initial speed of an object that comes to rest due to friction, use the work-energy theorem and the definition of frictional force.

Momentum and Collisions

• Impulse: The impulse delivered to an object is the product of the average force and the time interval during which the force acts.

• Collisions: In perfectly inelastic collisions, objects stick together after collision. Use conservation of momentum to find final velocities.

• Kinetic Energy Loss: In inelastic collisions, some kinetic energy is transformed into other forms (e.g., heat, sound).

Representative Table: Conservation Laws in Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls colliding
Inelastic	Yes	No	Car crash with cars sticking together
Perfectly Inelastic	Yes	No	Clay balls sticking together

Examples and Applications

• Pendulum: At the lowest point, all energy is kinetic; at the highest, all is potential.

• Roller Coaster: The car's speed is greatest at the lowest point due to conversion of potential energy to kinetic energy.

• Projectile: At the peak, velocity is minimum and potential energy is maximum.

• Spring Compression: The energy stored in a compressed spring is given by $\frac{1}{2}kx^2$.

Summary of Key Formulas

• $W = Fd \cos \theta$

• $KE = \frac{1}{2}mv^2$

• $PE_g = mgh$

• $PE_{\text{elastic}} = \frac{1}{2}kx^2$

• $ME = KE + PE$

• $P = \frac{W}{\Delta t}$

• $\vec{p} = m\vec{v}$

• $\vec{J} = \Delta \vec{p}$

• $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Additional info: These notes cover topics from chapters on Work & Energy, Conservation of Energy, Momentum & Impulse, and Collisions, as outlined in a typical college physics curriculum.