PHY2048 Exam 1 Study Guide: Chapters 1–5 (Vectors, Kinematics, Newton’s Laws, and Applications)

Vectors and Their Addition

Addition of Vectors by Components

Vectors are quantities that have both magnitude and direction. In physics, many quantities such as displacement, velocity, and force are vectors. Adding vectors by components is a systematic method that simplifies calculations, especially when vectors are not aligned along the same direction.

• Component Method: Break each vector into its x- and y-components using trigonometric functions.

• Add Corresponding Components: Sum all x-components to get the resultant x-component, and all y-components to get the resultant y-component.

• Resultant Vector: The resultant vector is found by combining the summed components.

Equations:

• For a vector \( \vec{A} \) at angle \( \theta \): $A_x = A \cos \theta$ $A_y = A \sin \theta$

• Resultant components: $R_x = \sum A_{x,i}$ $R_y = \sum A_{y,i}$

• Magnitude and direction of resultant: $R = \sqrt{R_x^2 + R_y^2}$ $\tan \phi = \frac{R_y}{R_x}$

Example: Add two vectors: 5 m at 30° and 8 m at 120°.

Motion Along a Straight Line (1D Kinematics)

1D Kinematic Variables

One-dimensional kinematics describes motion along a straight line using variables such as displacement, velocity, and acceleration.

• Displacement (\( \Delta x \)): Change in position.

• Velocity (\( v \)): Rate of change of displacement.

• Acceleration (\( a \)): Rate of change of velocity.

Equations of Motion (constant acceleration):

• $v = v_0 + a t$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

Free Fall

Free fall refers to motion under the influence of gravity alone, with acceleration \( a = -g \) (downward, where \( g \approx 9.8 \; \mathrm{m/s^2} \)).

• All objects in free fall near Earth's surface accelerate downward at the same rate, regardless of mass (neglecting air resistance).

• Use kinematic equations with \( a = -g \).

Example: Dropping a ball from rest: $v = -gt$, $y = y_0 - \frac{1}{2}gt^2$

Graphical Analysis of 1D Kinematic Variables

Graphs are powerful tools for visualizing motion. Common graphs include position vs. time, velocity vs. time, and acceleration vs. time.

• Position-Time Graph: Slope gives velocity.

• Velocity-Time Graph: Slope gives acceleration; area under curve gives displacement.

• Acceleration-Time Graph: Area under curve gives change in velocity.

Example: A straight line on a velocity-time graph indicates constant acceleration.

Motion in Two or Three Dimensions

Projectile Motion

Projectile motion involves objects moving in two dimensions under the influence of gravity. The horizontal and vertical motions are independent except for the time of flight.

• Horizontal Motion: Constant velocity (no horizontal acceleration).

• Vertical Motion: Constant acceleration due to gravity.

Equations:

• Horizontal: $x = v_{0x} t$

• Vertical: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• Initial velocity components: $v_{0x} = v_0 \cos \theta$ $v_{0y} = v_0 \sin \theta$

Example: A ball is launched at 20 m/s at 30° above the horizontal. Find its range and maximum height.

Uniform Circular Motion

Uniform circular motion describes an object moving in a circle at constant speed. The velocity changes direction, so there is a centripetal acceleration directed toward the center of the circle.

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

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

Example: A car rounds a curve of radius 50 m at 10 m/s. Find the required centripetal force.

Newton’s Laws of Motion and Applications

Newton’s Laws on an Elevator

Elevator problems involve analyzing forces when an object is in a non-inertial frame (accelerating up or down). The apparent weight changes depending on the acceleration.

• Apparent Weight: $N = m(g + a)$ (upward acceleration), $N = m(g - a)$ (downward acceleration)

• Free-Body Diagram: Draw all forces acting on the object (gravity, normal force).

Example: A 70 kg person in an elevator accelerating upward at 2 m/s²: $N = 70(9.8 + 2) = 826$ N

Multi-Object Accelerating Systems

Systems with multiple objects connected (e.g., by ropes or pulleys) require analyzing the forces on each object and applying Newton’s Second Law to the system.

• Draw Free-Body Diagrams: For each object, identify all forces.

• Write Equations: Apply $\sum F = m a$ for each object.

• Constraint: Objects connected by a rope have the same magnitude of acceleration.

Example: Two blocks connected by a string over a pulley; find acceleration and tension.

Multi-Object Systems in Equilibrium or Accelerating

Equilibrium means net force is zero; for accelerating systems, net force equals mass times acceleration. Analyze each object and the system as a whole.

• Equilibrium: $\sum F = 0$

• Accelerating: $\sum F = m a$

Example: Two blocks on a table, one hanging off the edge, connected by a string.

Newton’s Second Law

Newton’s Second Law relates the net force on an object to its acceleration.

• $\sum \vec{F} = m \vec{a}$

• Acceleration is in the direction of the net force.

Example: A 5 kg object with a net force of 20 N: $a = \frac{20}{5} = 4$ m/s²

Slanted Surfaces (Inclined Planes)

Without Friction

• Decompose weight into components parallel and perpendicular to the incline.

• Parallel component: $mg \sin \theta$

• Perpendicular component: $mg \cos \theta$

• Acceleration down the incline: $a = g \sin \theta$

With Friction

• Frictional force: $f = \mu N$, where $N = mg \cos \theta$

• Net force down the incline: $mg \sin \theta - f$

• Acceleration: $a = g \sin \theta - \mu g \cos \theta$

Example: A block slides down a 30° incline with $\mu = 0.2$.

Static and Kinetic Friction

Friction opposes relative motion between surfaces. Static friction prevents motion; kinetic friction acts during motion.

• Static Friction: $f_s \leq \mu_s N$ (maximum value before motion starts)

• Kinetic Friction: $f_k = \mu_k N$ (constant during motion)

• Usually, $\mu_s > \mu_k$

Example: A box requires a force of 50 N to start moving (static), but only 40 N to keep moving (kinetic).

Additional info: This guide expands on the listed exam topics with academic context, definitions, and representative equations to provide a comprehensive review for PHY2048 Exam 1 (Chapters 1–5).