Exam #3 Study Guide: Linear Momentum, Rotational Motion, and Fluids

Exam #3 Study Guide: Linear Momentum, Rotational Motion, and Fluids

Chapter 9: Linear Momentum and Collisions

This chapter covers the fundamental concepts of momentum, impulse, and collisions, including graphical analysis and center of mass calculations.

• Impulse: Impulse is the change in momentum of an object when a force is applied over a period of time. It is given by the equation: $\text{Impulse} = F \Delta t = \Delta p$

• Momentum: Momentum is the product of an object's mass and velocity. It is a vector quantity. $p = m v$

• Force vs. Time Graphs: The area under a force vs. time graph represents the impulse delivered to an object.

• Collisions: Collisions can be classified as elastic or inelastic. Conservation of momentum applies to all collisions. $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Center of Mass: The center of mass is the weighted average position of all the mass in a system. $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

• Example: A two-object collision where both masses and velocities are known; use conservation of momentum to solve for final velocities.

Chapter 10: Rotational Kinematics and Energy

This chapter introduces angular motion, the relationship between linear and angular quantities, and rotational energy concepts.

• Angular Kinematics Equations: Analogous to linear kinematics, these equations describe angular displacement, velocity, and acceleration. $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ $\omega = \omega_0 + \alpha t$ $\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)$

• Converting Between Linear and Angular Quantities: The relationship between linear and angular variables is: $v = r \omega$ $a = r \alpha$

• Moment of Inertia: Moment of inertia quantifies an object's resistance to rotational acceleration. $I = \sum m_i r_i^2$ Example values: For a solid disk: $I = \frac{1}{2} m r^2$; for a thin rod about its center: $I = \frac{1}{12} m L^2$.

• Rotational Kinetic Energy: The energy associated with rotating objects. $K_{rot} = \frac{1}{2} I \omega^2$

• Example: Calculating the rotational kinetic energy of a spinning disk.

Chapter 11: Rotational Dynamics and Static Equilibrium

This chapter explores torque, equilibrium, Newton's Second Law for rotation, and angular momentum.

• Torque: Torque is the rotational equivalent of force, causing angular acceleration. $\tau = r F \sin \theta$

• Balance (Equilibrium) Scenarios: An object is in equilibrium if the net force and net torque are both zero. $\sum F = 0$ $\sum \tau = 0$

• Newton's Second Law for Rotations: The angular analog of Newton's Second Law. $\sum \tau = I \alpha$

• Angular Momentum: Angular momentum is conserved in the absence of external torques. $L = I \omega$ $\sum \tau = \frac{dL}{dt}$

• Example: A seesaw in balance, or a spinning figure skater pulling in arms to spin faster.

Chapter 15: Fluids

This chapter covers properties of fluids, including density, pressure, buoyancy, and fluid motion.

• Density: Density is mass per unit volume. $\rho = \frac{m}{V}$

• Pressure with Depth: Pressure increases with depth in a fluid. $P = P_0 + \rho g h$

• Buoyant Force: The upward force exerted by a fluid on a submerged object. $F_{B} = \rho_{fluid} V_{displaced} g$

• Sinking vs. Floating: An object floats if its average density is less than the fluid; sinks if greater.

• Moving Fluids: The continuity equation and Bernoulli's equation describe fluid flow. Continuity: $A_1 v_1 = A_2 v_2$ Bernoulli's: $P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$

• Example: Calculating the buoyant force on a submerged block or the speed of water exiting a pipe.

Exam Structure and Preparation Tips

• Question Types:

  • Conceptual questions (multiple choice, short answer): 24% of exam grade

  • Short numerical problems: 51% of exam grade

  • Longer numerical problems: 25% of exam grade

• Key Problem Types:

  • Impulse and collisions (including vector components)

  • Angular kinematics and rotational energy

  • Newton's Second Law for rotations and angular momentum

  • Buoyancy and moving fluids

  • Balanced forces and torques

• Preparation Strategies:

  • Practice problems, especially new ones

  • Review equation sheet and variable meanings

  • Use online quizzes for concept review

• Exam Materials: Bring a calculator (non-graphing), writing tools, and be prepared to receive an equation sheet.

Summary Table: Key Equations and Concepts

Topic	Key Equation	Concept
Impulse	$\text{Impulse} = F \Delta t = \Delta p$	Change in momentum
Momentum	$p = m v$	Mass times velocity
Angular Kinematics	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$	Angular displacement
Moment of Inertia	$I = \sum m_i r_i^2$	Rotational mass distribution
Rotational Kinetic Energy	$K_{rot} = \frac{1}{2} I \omega^2$	Energy of rotation
Torque	$\tau = r F \sin \theta$	Rotational force
Density	$\rho = \frac{m}{V}$	Mass per unit volume
Pressure with Depth	$P = P_0 + \rho g h$	Fluid pressure
Buoyant Force	$F_{B} = \rho_{fluid} V_{displaced} g$	Upward force in fluid
Continuity Equation	$A_1 v_1 = A_2 v_2$	Conservation of flow
Bernoulli's Equation	$P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$	Energy in moving fluid

Additional info: Academic context and expanded explanations have been added to ensure completeness and clarity for exam preparation.