Physics 7A Final Exam Study Guide: Classical Mechanics, Fluids, and Oscillations

Classical Mechanics: Kinematics and Dynamics

One- and Two-Dimensional Kinematics

Kinematics describes the motion of objects without reference to the forces causing the motion. It includes both one-dimensional and two-dimensional motion.

• Displacement, Velocity, and Acceleration: Displacement is the change in position, velocity is the rate of change of displacement, and acceleration is the rate of change of velocity.

• Equations of Motion: For constant acceleration, the following equations are used:

  • $x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v(t) = v_0 + a t$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Projectile Motion: In two dimensions, motion is analyzed separately in the x and y directions. The horizontal motion has constant velocity, while the vertical motion has constant acceleration due to gravity.

• Example: A satellite in elliptical orbit around Earth involves both radial and tangential components of velocity and acceleration.

Forces and Newton's Laws

Newton's laws of motion describe the relationship between the motion of an object and the forces acting on it.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: The net force on an object is equal to the mass times its acceleration: $\vec{F}_{net} = m \vec{a}$

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• Example: The tension in a wire and gravitational forces in a torsion balance experiment.

Circular Motion and Gravitation

Circular Motion

Objects moving in a circle experience a centripetal acceleration directed toward the center of the circle.

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

• Centripetal Force: $F_c = m \frac{v^2}{r}$

• Example: Satellite motion around Earth, including torque and angular momentum considerations.

Universal Gravitation

Newton's law of universal gravitation describes the attractive force between two masses.

• Gravitational Force: $F = G \frac{m_1 m_2}{r^2}$

• Gravitational Constant: $G$ is determined experimentally, such as with a torsion balance.

• Example: Calculation of $G$ using the torsion balance and oscillation period.

Work, Energy, and Power

Work and Energy

Work is done when a force causes displacement. Energy is the capacity to do work.

• Work: $W = F d \cos \theta$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy: $U = mgh$ (gravitational), $U = \frac{1}{2} k x^2$ (spring)

• Conservation of Energy: Total mechanical energy is conserved in the absence of non-conservative forces.

• Example: Fluid flow and pressure differences in a tube, energy transformations in oscillators.

Momentum and Collisions

Linear Momentum

Momentum is the product of mass and velocity. In a closed system, total momentum is conserved.

• Momentum: $p = m v$

• Impulse: $J = F \Delta t = \Delta p$

• Conservation of Momentum: $\sum p_{initial} = \sum p_{final}$

• Example: Collisions between blocks in oscillation problems.

Rotational Dynamics and Angular Momentum

Rotational Kinematics and Dynamics

Rotational motion involves angular displacement, velocity, and acceleration.

• Angular Displacement: $\theta$ (radians)

• Angular Velocity: $\omega = \frac{d\theta}{dt}$

• Angular Acceleration: $\alpha = \frac{d\omega}{dt}$

• Rotational Inertia (Moment of Inertia): $I = \sum m_i r_i^2$ for discrete masses, or $I = \int r^2 dm$ for continuous distributions.

• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$

• Torque: $\tau = r F \sin \theta$

• Angular Momentum: $L = I \omega$

• Conservation of Angular Momentum: $L_{initial} = L_{final}$

• Example: Disk rolling down a track, calculation of speed and height using energy and angular momentum conservation.

Rotational Inertia Table

The moment of inertia depends on the mass distribution relative to the axis of rotation.

Object	Moment of Inertia ($I$)
Hoop or cylindrical shell	$I = M R^2$
Hollow cylinder or disk	$I = M R^2$
Solid cylinder or disk	$I = \frac{1}{2} M R^2$
Rectangular plate	$I = \frac{1}{12} M L^2$
Long thin rod (center)	$I = \frac{1}{12} M L^2$
Long thin rod (end)	$I = \frac{1}{3} M L^2$
Solid sphere	$I = \frac{2}{5} M R^2$
Thin spherical shell	$I = \frac{2}{3} M R^2$

Fluid Dynamics

Fluid Flow and Pressure

Fluid dynamics studies the motion of fluids and the forces acting on them.

• Continuity Equation: $A_1 v_1 = A_2 v_2$ (for incompressible fluids)

• Bernoulli's Equation: $P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$

• Pressure Difference: $\Delta P = \rho g h$ (static fluids)

• Example: Determining flow velocity and pressure differences in a manometer tube.

Simple Harmonic Motion

Oscillators and Standing Waves

Simple harmonic motion (SHM) describes systems where the restoring force is proportional to displacement.

• Equation of Motion: $x(t) = A \cos(\omega t + \phi)$

• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$

• Period: $T = \frac{2\pi}{\omega}$

• Standing Waves: Formed by the superposition of two waves traveling in opposite directions. The boundary conditions (open or closed ends) determine the possible frequencies.

• Example: Determining the frequency and boundary conditions of a tube based on pressure nodes and antinodes.

Waves and Sound

Wave Properties

Waves transfer energy through oscillations. Sound waves are longitudinal waves in a medium.

• Wave Speed: $v = f \lambda$

• Standing Waves in Tubes: The frequency depends on whether the tube is open or closed at the ends.

• Example: Analyzing pressure variation graphs to determine tube boundary conditions and frequencies.

Additional Mathematical Tools

Useful Equations and Integrals

• Trigonometric Identities: Used in wave and oscillation problems.

• Integrals: For calculating areas, volumes, and solving equations of motion.

• Taylor Expansion: $f(x) = f(a) + f'(a)(x-a) + \frac{1}{2} f''(a)(x-a)^2 + \ldots$

Summary Table: Key Equations

Topic	Equation
Kinematics	$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$
Newton's 2nd Law	$\vec{F}_{net} = m \vec{a}$
Work	$W = F d \cos \theta$
Kinetic Energy	$K = \frac{1}{2} m v^2$
Potential Energy	$U = mgh$
Momentum	$p = m v$
Impulse	$J = F \Delta t$
Rotational Inertia	$I = \sum m_i r_i^2$
Torque	$\tau = r F \sin \theta$
Angular Momentum	$L = I \omega$
Bernoulli's Equation	$P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$
SHM Frequency	$\omega = \sqrt{\frac{k}{m}}$

Exam Preparation Tips

• Understand the physical meaning behind each equation.

• Practice applying equations to different scenarios, such as collisions, oscillations, and fluid flow.

• Review the boundary conditions for standing waves and rotational inertia for various shapes.

• Use the provided equation sheet for quick reference during problem solving.

Additional info: The study guide covers topics from Ch.2 (Kinematics), Ch.4 (Forces and Motion), Ch.5 (Circular Motion and Gravitation), Ch.6 (Energy, Work, and Power), Ch.7 (Momentum and Collisions), Ch.8 (Rotational Dynamics), Ch.9 (Fluid Dynamics), Ch.10 (Simple Harmonic Motion), and Ch.11 (Waves and Sound), as indicated by the exam questions and equation sheet.