PHYS 1403: Final Exam Practice Problems and Key Concepts

Ch1: Introduction to Physics

Fundamental Concepts

Physics is the study of the fundamental laws of nature, describing how matter and energy interact. It relies on precise measurement and analysis to understand the physical world.

• SI Units: The International System of Units (SI) is the standard for scientific measurements. Key base units include meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

• Unit Conversions: Converting between units is essential for solving physics problems. Use conversion factors to change from one unit to another.

• Uncertainties & Significant Figures: Measurements have inherent uncertainties. Significant figures reflect the precision of a measurement.

• Dimensional Analysis: A method to check the consistency of equations and convert units by analyzing the dimensions (e.g., length, mass, time).

• Problem Solving Strategy: Systematic approaches include identifying knowns/unknowns, drawing diagrams, and applying relevant equations.

Ch2: 1D Kinematics

Motion Along a Straight Line

Kinematics describes the motion of objects without considering the forces that cause the motion.

• Position, Displacement, Distance: Position is the location of an object. Displacement is the change in position ($\Delta x$), and distance is the total path length traveled.

• Velocity: Average velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Instantaneous velocity: The velocity at a specific moment.

• Acceleration: $a = \frac{\Delta v}{\Delta t}$

• Graphical Analysis: Position-time ($x$-$t$) and velocity-time ($v$-$t$) graphs help visualize motion.

• Equations of Motion (Constant Acceleration): $\begin{align*} x &= x_0 + v_0 t + \frac{1}{2} a t^2 \\ v &= v_0 + a t \\ v^2 &= v_0^2 + 2a(x - x_0) \end{align*}$

• Free Fall: Objects under gravity accelerate at $g \approx 9.8\,\text{m/s}^2$ downward.

Ch3: 2D Kinematics

Vectors and Motion in a Plane

Two-dimensional motion requires vector analysis to describe position, velocity, and acceleration.

• Vectors and Scalars: Scalars have magnitude only; vectors have both magnitude and direction.

• Operations with Vectors: Graphical methods (tip-to-tail) and component methods (using $x$ and $y$ axes).

• Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Motion in 2 Dimensions: Analyze $x$ and $y$ motions separately, often under constant acceleration.

• Projectile Motion: Horizontal and vertical motions are independent. $\begin{align*} x(t) &= v_{0x} t \\ y(t) &= v_{0y} t - \frac{1}{2} g t^2 \end{align*}$

Ch4: Force and Motion I: Newton's Laws

Fundamental Laws of Motion

Newton's laws describe the relationship between forces and the motion of objects.

• Forces: A force is a push or pull acting on an object.

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

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

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

• Types of Forces: Normal force (support), weight ($mg$), tension (in ropes/cables).

• Free Body Diagram: A diagram showing all forces acting on an object.

Ch5: Force and Motion II: Applications

Frictional Forces

Friction opposes motion between surfaces in contact.

• Static Friction: Prevents motion up to a maximum value: $f_s \leq \mu_s N$

• Kinetic Friction: Acts during motion: $f_k = \mu_k N$

Ch6: Work and Energy

Energy Transfer and Conservation

Work and energy concepts explain how forces cause changes in motion and energy.

• Work: $W = F d \cos \theta$

• Work-Energy Theorem: $W_{net} = \Delta K = K_f - K_i$

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

• Hooke's Law: $F = -k x$ (spring force)

• Work by Varying Force: $W = \int F(x) dx$

• Potential Energy: Gravitational: $U_g = mgh$ Spring: $U_s = \frac{1}{2} k x^2$

• Conservative vs. Non-Conservative Forces: Conservative forces (e.g., gravity, springs) conserve mechanical energy; non-conservative (e.g., friction) do not.

• Conservation of Energy: $E_{total} = K + U$ remains constant for isolated systems.

• Power: $P = \frac{dW}{dt}$

Ch8: Momentum and Collisions

Linear Momentum and Impulse

Momentum is a measure of motion, and its conservation is fundamental in collisions.

• Linear Momentum: $\vec{p} = m \vec{v}$

• Conservation of Momentum: In a closed system, total momentum before and after a collision is constant.

• Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Only momentum is conserved; kinetic energy is not.

• Impulse: $\vec{J} = \Delta \vec{p} = F_{avg} \Delta t$

Ch9: Rotational Motion

Rotational Kinematics and Dynamics

Rotational motion involves angular quantities analogous to linear motion.

• Angular Displacement, Velocity, Acceleration: $\theta$, $\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$

• Relation to Linear Quantities: $v = r \omega$, $a = r \alpha$

• Equations of Rotational Motion: $\begin{align*} \theta &= \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \\ \omega &= \omega_0 + \alpha t \\ \omega^2 &= \omega_0^2 + 2\alpha(\theta - \theta_0) \end{align*}$

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

• Torque: $\tau = r F \sin \theta$

• Moment of Inertia: $I = \sum m_i r_i^2$

• Rotational Form of Newton's Second Law: $\tau_{net} = I \alpha$

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

• Angular Momentum: $L = I \omega$

• Rotational Collisions: Conservation of angular momentum applies when no external torque acts.

Example: Rolling Cylinder Up a Hill

A solid cylinder (moment of inertia $I = \frac{1}{2} m R^2$) rolls uphill with initial speed. To determine if it reaches the top, use conservation of energy:

• Initial energy: $E_i = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

• Final energy at top: $E_f = mgh$

• If $E_i < E_f$, the cylinder cannot reach the top.

Answer: No, the cylinder does not reach the top if its initial energy is insufficient.

Example: Rotational Collision of Discs

Two discs stick together after spinning in opposite directions. Use conservation of angular momentum:

• $L_{initial} = I_A \omega_A + I_B \omega_B$

• $L_{final} = (I_A + I_B) \omega_{final}$

• Solve for $\omega_{final}$: $\omega_{final} = \frac{I_A \omega_A + I_B \omega_B}{I_A + I_B}$

Answer: $-1/3$ rad/s (CCW is positive)

Ch13: Fluid Mechanics

Properties of Fluids

Fluid mechanics studies the behavior of liquids and gases under various conditions.

• Density: $\rho = \frac{m}{V}$

• Pressure: $P = \frac{F}{A}$

• Variation of Pressure with Depth: $P = P_0 + \rho g h$

• Buoyancy: The upward force exerted by a fluid on a submerged object.

• Archimedes' Principle: The buoyant force equals the weight of the fluid displaced: $F_b = \rho_{fluid} V_{displaced} g$

Example: Maximum Pressure by a Brick

Maximum pressure is exerted when the brick rests on its smallest face:

• Area: $A = 0.3 \times 0.1 = 0.03\,\text{m}^2$

• Pressure: $P = \frac{F}{A}$

Answer: $12.5$ kPa

Example: Pressure on an Immersed Brick

Pressure increases with depth. The greatest pressure is at the bottom face of the brick.

Example: Water Pressure in a Straw

Pressure at the top of a straw of length $l$:

• $P_{top} = P_{atm} - \rho g l$

Answer: $2.26 \times 10^4$ Pa

Example: Buoyant Force Comparison

Buoyant force depends on the volume of fluid displaced. For equal masses, the less dense material (aluminum) displaces more water, thus experiences a greater buoyant force.

Example: Pressure in a U-Tube

At any given horizontal level in a connected fluid, the pressure is the same on both sides.

Concept	Equation	Key Points
Density	$\rho = \frac{m}{V}$	Mass per unit volume
Pressure	$P = \frac{F}{A}$	Force per unit area
Buoyant Force	$F_b = \rho_{fluid} V_{displaced} g$	Upward force on submerged object
Pressure with Depth	$P = P_0 + \rho g h$	Increases with depth

Additional info: Some equations and explanations have been expanded for clarity and completeness. All examples are based on the provided practice problems and standard physics principles.