Comprehensive Study Notes for PHY250 Physics I Final Exam Topics

Rotational Motion and Moment of Inertia

Moment of Inertia for Common Objects

The moment of inertia quantifies an object's resistance to changes in rotational motion about a specific axis. It depends on the mass distribution relative to the axis of rotation.

• Solid Sphere: $I = \frac{2}{5}MR^2$

• Hollow Sphere: $I = \frac{2}{3}MR^2$

• Solid Cylinder (axis through center): $I = \frac{1}{2}MR^2$

• Thin Rod (axis through center): $I = \frac{1}{12}ML^2$

• Thin Rod (axis through end): $I = \frac{1}{3}ML^2$

Example: Calculating the moment of inertia for a solid sphere of mass 2 kg and radius 0.5 m:

• $I = \frac{2}{5} \times 2 \times (0.5)^2 = 0.2 \text{ kg} \cdot \text{m}^2$

Motion in a Straight Line and Two Dimensions

Displacement, Velocity, and Vector Addition

Displacement is a vector quantity representing the change in position. When combining movements in different directions, use vector addition.

• Displacement Vector: $\vec{\Delta x} = \vec{x}_2 - \vec{x}_1$

• Magnitude: $|\vec{\Delta x}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Direction: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Example: Walking 150 m east and then 220 m north:

• Magnitude: $\sqrt{150^2 + 220^2} = 266.27 \text{ m}$

• Direction: $\theta = \tan^{-1}\left(\frac{150}{220}\right) = 34^\circ$ north of east

Force and Motion

Newton's Laws and Forces in Equilibrium

Newton's laws describe the relationship between forces and motion. In equilibrium, the sum of forces is zero.

• Newton's Second Law: $\sum F = ma$

• Equilibrium: $\sum F = 0$

• Example (Parachutist): At constant velocity, $F_{drag} = mg$

Tension in Ropes

For objects suspended by ropes, tension can be found using equilibrium conditions.

• $T = \frac{mg}{2 \cos \theta}$ (for a mass hanging at the center of two ropes at angle $\theta$)

Work, Energy, and Power

Calculating Power Output

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

• $P = \frac{\text{Work}}{\text{Time}}$

• Example: $P = \frac{22.4 \times 10^3}{10.6} = 2.11 \text{ kW}$

Gravity and Circular Orbits

Satellite Orbits and Altitude Calculation

Satellites in circular orbits around Earth obey gravitational and centripetal force relationships.

• Orbital Period: $T = 2\pi \sqrt{\frac{(R+h)^3}{GM_E}}$

• Altitude for Given Period: $h = \sqrt[3]{\frac{GMT^2}{4\pi^2}} - R$

Conservation of Momentum

Collisions and Momentum Conservation

In collisions, total momentum before equals total momentum after.

• $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

• Example: Two masses collide and move apart; use conservation to solve for final velocities.

Rotational Motion and Angular Speed

Angular Speed Calculations

Angular speed ($\omega$) is the rate of rotation, given by:

• $\omega = \frac{2\pi}{T}$

• Example: Earth's angular speed: $\omega = \frac{2\pi}{86400} = 7.3 \times 10^{-5} \text{ rad/s}$

Fluid Motion and Buoyancy

Density and Buoyancy

Objects float or sink depending on their density relative to the fluid.

• Density: $\rho = \frac{m}{V}$

• Buoyant Force: $F_B = \rho_{fluid} V_{object} g$

• Normal Force from Bottom: $F_N = mg - F_B$

Oscillatory and Wave Motion

Wave Properties and Energy

Periodic waves have frequency, amplitude, and energy. Energy is proportional to frequency.

• Frequency: $f = \frac{1}{T}$

• Energy Ratio: $\frac{E_a}{E_b} = \frac{f_a}{f_b}$

Sound Intensity and Decibel Level

Sound intensity ($I$) and decibel level ($\beta$) are related by:

• $I = \frac{P}{4\pi r^2}$

• $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I_0 = 10^{-12} \text{ W/m}^2$

Summary Table: Key Equations and Concepts

Topic	Key Equation	Application
Displacement	$|\vec{\Delta x}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Finding total movement
Newton's Second Law	$\sum F = ma$	Force and acceleration
Power	$P = \frac{\text{Work}}{\text{Time}}$	Rate of energy transfer
Orbital Period	$T = 2\pi \sqrt{\frac{(R+h)^3}{GM_E}}$	Satellite orbits
Momentum Conservation	$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$	Collisions
Angular Speed	$\omega = \frac{2\pi}{T}$	Rotational motion
Density	$\rho = \frac{m}{V}$	Buoyancy
Sound Intensity	$I = \frac{P}{4\pi r^2}$	Wave energy
Decibel Level	$\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$	Sound measurement

Additional info: These notes synthesize sample final exam questions and solutions, covering core topics from introductory college physics including kinematics, dynamics, energy, rotation, gravity, fluids, and waves. All equations are presented in LaTeX for clarity and academic rigor.