Chapter 04

Course Logistics and Announcements

Course Communication and Support

• iClicker, lecture notes, office hours: Contact your instructor for section-specific issues.

• Recitations: Contact your TA or coordinators.

• Grades, absences, logistics: Contact course administrators (Drs. May and Youngs).

• Technical issues with Mastering: Contact Pearson support.

Exam and Assignment Reminders

• Exam covers up to Chapter 3: Math, units, and kinematics (2D and 3D).

• Assignments due Saturday at 11:59 pm; keep up with coursework.

• Chapter 5 pre-lecture due the morning of the exam.

Kinematics Review

Constant Speed and Acceleration

• Constant speed: If a car moves at a constant speed (e.g., 15 mph), acceleration is zero because velocity does not change in magnitude or direction.

• Acceleration: Change in velocity over time. If direction changes (e.g., turning), acceleration may be present even if speed is constant.

Average Acceleration Calculation

• Given initial velocity $\vec{v}_0$ and final velocity $\vec{v}$ over time interval $\Delta t$, average acceleration is:

$\vec{a}_{\text{avg}} = \frac{\vec{v} - \vec{v}_0}{\Delta t}$

• Example: If $\vec{v}_0 = -5\,\text{m/s}$ and $\vec{v} = (4t - 3)\,\text{m/s}$ at $t = 1\,\text{s}$, calculate $\vec{a}_{\text{avg}}$:

$\Delta t = 1\,\text{s} \qquad a_x = 4\,\text{m/s}^2 \qquad a_y = 2\,\text{m/s}^2$

$|\vec{a}| = \sqrt{a_x^2 + a_y^2} = \sqrt{4^2 + 2^2} = 4.5\,\text{m/s}^2$

Projectile Motion: Speed vs. Time Graphs

• For a stone thrown at an angle with negligible air resistance, the speed vs. time graph is symmetric and parabolic, decreasing as the stone rises, reaching a minimum at the top, then increasing as it falls.

• Correct graph: Option (d) (parabolic shape).

Relative Velocity

• Relative velocity: The velocity of one object as observed from another moving object.

• Example: If Alice moves at $15\,\text{m/s}$ relative to Bob, and the boat moves at $10\,\text{m/s}$, Alice's velocity as seen by her dog on the boat is $+5\,\text{m/s}$.

Forces and Newton's Laws

Introduction to Forces

• Force: A push or pull acting upon an object resulting from its interaction with another object.

• Forces are vectors (they have magnitude and direction).

• Types of forces include gravitational, electromagnetic, strong nuclear, and weak nuclear (the four fundamental forces).

Combining Forces: Superposition Principle

• The net force on an object is the vector sum of all individual forces acting on it.

$\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \ldots$

• In two or three dimensions, add components separately:

$F_{\text{net},x} = F_{1,x} + F_{2,x} + \ldots$

$F_{\text{net},y} = F_{1,y} + F_{2,y} + \ldots$

Newton's First Law (Law of Inertia)

• An object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.

• Inertia: The tendency of an object to resist changes in its state of motion; quantitatively measured by mass.

Newton's Second Law

• The net force acting on an object is equal to the mass of the object multiplied by its acceleration.

$\vec{F}_{\text{net}} = m\vec{a}$

• Component form:

$F_x = m a_x \qquad F_y = m a_y \qquad F_z = m a_z$

• Example: If two pucks of different mass are pushed with the same force, the lighter puck will accelerate more and travel farther in the same time.

Newton's Third Law

• For every action, there is an equal and opposite reaction.

• If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.

• Action-reaction pairs act on different objects, not on the same object.

Free-Body Diagrams (FBDs)

• A free-body diagram is a vector diagram showing all forces acting on a single object.

• Helps in setting up equations using $\sum \vec{F} = m\vec{a}$.

• Do not include forces the object exerts on others; only forces acting on the object itself.

Mass vs. Weight

Definitions and Differences

• Mass: A measure of the amount of matter in an object; scalar quantity measured in kilograms (kg).

• Weight: The force of gravity acting on an object; vector quantity measured in newtons (N).

$W = mg$

• On Earth, $g \approx 9.8\,\text{m/s}^2$; on other planets, $g$ varies.

• Example: On Pluto, $g \approx 0.54\,\text{m/s}^2$.

Sample Problems and Applications

Force Components on an Inclined Plane

• To find the force $|F|$ needed for a component parallel to the ramp ($F_{\parallel}$):

$F_{\parallel} = F \cos \theta$

$F = \frac{F_{\parallel}}{\cos \theta}$

• Example: If $F_{\parallel} = 60\,\text{N}$ and $\theta = 30^\circ$, $F = 60 / \cos 30^\circ$.

Friction and Motion

• When a force is applied to an object on a surface with friction, the distance traveled depends on initial velocity, frictional force, and mass.

• Work-energy principle can be used:

$W_{\text{friction}} = F_{\text{friction}} \cdot d = \frac{1}{2} m v_0^2$

• Solve for $d$ (distance):

$d = \frac{\frac{1}{2} m v_0^2}{F_{\text{friction}}}$

Elevator Example

• When standing on a scale in an accelerating elevator, the scale reads the normal force, which may differ from true weight.

• If elevator accelerates upward:

$N = m(g + a)$

• If elevator accelerates downward:

$N = m(g - a)$

Summary Table: Newton's Laws

Law	Statement	Key Equation	Example
First Law	Object remains at rest or in uniform motion unless acted on by net force	$\sum \vec{F} = 0$	Book resting on a table
Second Law	Net force equals mass times acceleration	$\sum \vec{F} = m\vec{a}$	Puck pushed across ice
Third Law	For every action, equal and opposite reaction	$\vec{F}_{A \to B} = -\vec{F}_{B \to A}$	Hand pushes wall, wall pushes back

Additional info:

• Some context and examples have been expanded for clarity and completeness.

• Diagrams referenced (e.g., FBDs, speed-time graphs) are described in text for self-contained study.