PH101 Unit Exam #1 Study Notes: Chapters 1-5, 7

Models, Measurements & Vectors

Units, Dimensions, and Vector Addition

Understanding units and vectors is fundamental in physics, as they form the basis for describing and analyzing physical quantities and their relationships.

• Units and Dimensions: Every physical quantity has associated units (e.g., meters, seconds, kilograms) and dimensions (e.g., length [L], time [T], mass [M]). Always write units next to every number in calculations to ensure clarity and correctness.

• Dimensional Analysis: Use the dimensions of physical quantities to check the validity of equations. For example, both sides of an equation must have the same dimensions.

• Vector Addition: Vectors have both magnitude and direction. To add two vectors in a North-South-East-West coordinate system, use graphical (tip-to-tail) or analytical (component-wise) methods.

• Expressing Vectors: The sum of two vectors can be expressed as a magnitude and a direction, often using trigonometry to find the resultant.

• Example: If vector A points North and vector B points East, the resultant vector R can be found using the Pythagorean theorem and the arctangent function for direction.

Formula:

• Magnitude of resultant: $R = \sqrt{A^2 + B^2}$

• Direction: $\theta = \arctan\left(\frac{B}{A}\right)$

Motion Along a Straight Line

1-D Kinematics

One-dimensional kinematics involves analyzing the motion of objects along a straight path, considering velocity, acceleration, and displacement.

• Velocity and Acceleration: The direction of initial velocity and acceleration determines whether an object speeds up or slows down.

• Free Fall and Thrown Objects: For objects thrown downward, use kinematic equations to relate initial speed, time, and displacement.

• Example: If an object is thrown downward from a building with initial speed $v_0$ and hits the ground after time $t$, the height $h$ of the building is:

$h = v_0 t + \frac{1}{2} g t^2$

Motion in a Plane

2-D Kinematics & Projectile Motion

Projectile motion describes the motion of objects launched into the air, subject only to gravity (neglecting air resistance).

• Definition: Projectile motion consists of independent horizontal and vertical motions, with constant horizontal velocity and constant vertical acceleration ($g$).

• Time of Flight: For an object launched at speed $v_0$ and angle $\theta$, the time in the air is:

$t = \frac{2 v_0 \sin \theta}{g}$

• Horizontal Launch: For an object projected horizontally from height $h$ with speed $v_0$, the final velocity just before impact combines horizontal and vertical components:

Vertical velocity: $v_y = g t$ Horizontal velocity: $v_x = v_0$ Magnitude: $v = \sqrt{v_x^2 + v_y^2}$

• Example: A ball rolls off a table 2 m high with speed 3 m/s. Time to hit ground: $t = \sqrt{\frac{2h}{g}}$; final velocity: $v = \sqrt{v_0^2 + (g t)^2}$

Newton's Laws of Motion

Forces and Applications

Newton's laws describe the relationship between forces and motion, forming the foundation for classical mechanics.

• Constant Speed on Incline: If an object slides down an incline at constant speed, the sum of all forces (net force) is zero, and the direction of forces balances along the incline.

• Half-Atwood Machine: Two objects connected by a rope over a pulley, with one on a frictionless table. The tension in the rope can be compared to the weight of the hanging object.

• Friction and Inclines: Given coefficients of friction ($\mu_s$, $\mu_k$) and mass, determine the critical angle at which an object starts to slide:

$\theta_{critical} = \arctan(\mu_s)$

• Example: A box on a truck bed will start to slide when the incline angle exceeds $\arctan(\mu_s)$.

Applications of Newton's Laws

Half-Atwood Machine and Friction

Applying Newton's laws to systems such as the half-Atwood machine and objects on inclined planes helps analyze tension, friction, and equilibrium.

• Tension vs. Weight: In a half-Atwood machine, the tension in the rope is less than the weight of the hanging mass if the system accelerates.

• Friction: The force of friction is $f = \mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal force.

Work & Energy

Work-Energy Theorem and Friction

The work-energy theorem relates the work done on an object to its change in kinetic energy. Friction is a non-conservative force that dissipates energy.

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

$W_{net} = \Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

• Comparing Work: To change a car's speed from $v_i$ to $v_f$, the required work is proportional to the difference in the squares of the speeds.

• Friction and Stopping Distance: If an object stops due to friction over distance $d$, the coefficient of friction can be found using energy principles:

$\mu = \frac{v_i^2}{2 g d}$

• Example: A car of mass $m$ stops from speed $v_i$ over distance $d$ due to friction. The work done by friction equals the initial kinetic energy.