Study Guide: Energy, Work, Momentum, and Rotational Motion (Chapters 9–12)

Energy, Work, Momentum, and Rotational Motion

Chapter 9: Energy

This chapter introduces the concept of energy in physics, focusing on mechanical energy and its conservation in various systems.

• Basic Energy Model: Energy is a scalar quantity that can be stored in various forms and transferred between objects or systems.

• Conservation of Mechanical Energy: In the absence of non-conservative forces (like friction), the total mechanical energy (kinetic + potential) of a system remains constant.

  • Mathematically: $K_f + U_f = K_i + U_i$ or $\Delta K + \Delta U = 0$

• Kinetic Energy (K): The energy of motion.

  • Formula: $K = \frac{1}{2} m v^2$

• Gravitational Potential Energy (Ug): Energy stored due to an object's position in a gravitational field.

  • Formula: $U_g = mgy$

• Elastic Potential Energy (Us): Energy stored in a stretched or compressed spring.

  • Formula: $U_s = \frac{1}{2} k (\Delta s)^2$

• Energy Diagrams: Graphical representations of potential energy as a function of position, useful for visualizing equilibrium points and motion.

• Hooke's Law: Describes the force exerted by a spring.

  • Formula: $F_x = -k (\Delta s)$

• Example: A mass attached to a spring oscillates back and forth, converting kinetic energy to elastic potential energy and vice versa, with total mechanical energy conserved (if no friction).

Chapter 10: Work

This chapter explores the concept of work, its relationship to energy, and how external forces can change a system's energy.

• Modified Energy Equation: When external forces or non-conservative forces (like friction) are present, the change in a system's energy equals the work done by these forces.

  • Formula: $W_{ext} = \Delta E_{sys} = \Delta K + \Delta U + \Delta E_{th}$

• Work-Kinetic Energy Theorem: The net work done on an object equals its change in kinetic energy.

  • Formula: $\Delta K = W_{net} = W_c + W_{diss} + W_{ext}$

• Definition of Work: Work is the product of the force component along the direction of displacement and the magnitude of that displacement.

  • For variable force: $W = \int F_x \, dx$

  • Work is the area under the force vs. position graph.

• Conservative Forces: Forces for which the work done is path-independent and can be associated with a potential energy function.

  • Relationship: $F_x = -\frac{dU}{dx}$

• Power: The rate at which work is done or energy is transferred.

  • Formula: $P = \frac{dE_{sys}}{dt}$

• Example: Lifting a box at constant speed requires work against gravity, increasing the box's gravitational potential energy.

Chapter 11: Impulse and Momentum

This chapter covers the concepts of momentum and impulse, and the principle of conservation of momentum in collisions and explosions.

• Momentum (p): A vector quantity defined as the product of mass and velocity.

  • Formula: $p_x = m v_x$

• Impulse: The change in momentum of an object, equal to the net force applied times the duration of application.

  • Formula: $J = \Delta p = F_{net} \Delta t$

  • For variable force: $J = \int F_{net} \, dt$

• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.

  • Applies to collisions and explosions.

• Types of Collisions:

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

  • Totally Inelastic: Objects stick together after collision; momentum is conserved, but kinetic energy is not.

• Example: Two ice skaters push off from each other and move in opposite directions; their total momentum before and after remains zero.

Chapter 12: Rigid Body Rotations

This chapter introduces rotational motion, including kinematics, dynamics, energy, and angular momentum for rigid bodies.

• Rotational Kinematics: Describes angular position ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$).

  • Key equations (for constant $\alpha$):

    • $\omega_f = \omega_i + \alpha t$

    • $\theta = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$

    • $\omega_f^2 = \omega_i^2 + 2\alpha (\theta_f - \theta_i)$

• Relating Angular and Linear Variables:

  • Tangential velocity: $v_t = r \omega$

  • Tangential acceleration: $a_t = r \alpha$

  • Centripetal acceleration: $a_c = \frac{v^2}{r} = r \omega^2$

• Rolling Motion: For objects rolling without slipping, $v_{cm} = R \omega$

• Moment of Inertia (I): A measure of an object's resistance to changes in rotational motion.

  • For discrete masses: $I = \sum m_i r_i^2$

• Rotational Kinetic Energy:

  • Formula: $K_{rot} = \frac{1}{2} I \omega^2$

• Conservation of Energy in Rotational Motion: Total energy includes both translational and rotational kinetic energy.

• Torque (\(\tau\)): The rotational equivalent of force, causing angular acceleration.

  • Formula: $\tau = r F \sin \theta$

  • Net torque: $\sum \tau = I \alpha$

• Equilibrium: For an object to be in equilibrium, both the net force and net torque must be zero.

  • Translational: $\sum F = 0$

  • Rotational: $\sum \tau = 0$

• Angular Momentum (L): The rotational analog of linear momentum.

  • Formula: $L = I \omega$

  • Conservation: If net external torque is zero, $L$ is conserved.

• Work Done by a Torque:

  • Formula: $W = \tau \Delta \theta$

• Example: A spinning figure skater pulls in her arms, reducing her moment of inertia and increasing her angular velocity to conserve angular momentum.

Table: Comparison of Linear and Rotational Quantities

Linear Quantity	Rotational Analog	Formula
Displacement ($x$)	Angular Displacement ($\theta$)	$\theta = x / r$
Velocity ($v$)	Angular Velocity ($\omega$)	$v = r \omega$
Acceleration ($a$)	Angular Acceleration ($\alpha$)	$a = r \alpha$
Mass ($m$)	Moment of Inertia ($I$)	$I = \sum m_i r_i^2$
Force ($F$)	Torque ($\tau$)	$\tau = r F \sin \theta$
Linear Momentum ($p$)	Angular Momentum ($L$)	$L = I \omega$
Kinetic Energy ($K$)	Rotational Kinetic Energy ($K_{rot}$)	$K_{rot} = \frac{1}{2} I \omega^2$

Additional info: Students are advised to review assigned homework and in-class problems for practice, as exam questions will be similar in style and content.