Newton

Cheatsheet Content

Newton's First Law (Law of Inertia) An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. Inertia: The resistance of any physical object to any change in its state of motion, including changes to its speed, direction, or state of rest. Mathematical Representation: If $\sum \vec{F} = 0$, then $\vec{v} = \text{constant}$ (or $\vec{a} = 0$). Key Concept: Defines force as that which changes the state of motion. Newton's Second Law (Law of Acceleration) The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the net force. Mathematical Representation: $\sum \vec{F} = m\vec{a}$ $\sum \vec{F}$: Net force (vector sum of all forces acting on the object) in Newtons (N) $m$: Mass of the object in kilograms (kg) $\vec{a}$: Acceleration of the object in meters per second squared ($\text{m/s}^2$) Units: $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$ Components: $\sum F_x = ma_x$ $\sum F_y = ma_y$ $\sum F_z = ma_z$ Impulse-Momentum Theorem (derived from 2nd Law): $\vec{J} = \Delta \vec{p}$ $\vec{J} = \int \vec{F} dt = \vec{F}_{\text{avg}} \Delta t$ (Impulse) $\Delta \vec{p} = m\vec{v}_f - m\vec{v}_i$ (Change in Momentum) Newton's Third Law (Law of Action-Reaction) For every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object. Mathematical Representation: $\vec{F}_{AB} = -\vec{F}_{BA}$ $\vec{F}_{AB}$: Force exerted by object A on object B $\vec{F}_{BA}$: Force exerted by object B on object A Key Characteristics: Forces always occur in pairs. Action-reaction pairs act on different objects. They are equal in magnitude and opposite in direction. They are always of the same type (e.g., gravitational, normal, tension). Common Misconception: Action-reaction forces do not cancel each other out because they act on different objects. Common Forces Force Type Description Formula / Notes Weight ($F_g$) Force of gravity on an object near Earth's surface. $F_g = mg$ (where $g \approx 9.8 \text{ m/s}^2$) Normal Force ($F_N$) Perpendicular force exerted by a surface on an object in contact with it. Varies; often $F_N = F_g$ on a horizontal surface without vertical acceleration. Tension ($T$) Force transmitted through a string, rope, cable, or wire when pulled tight. Acts along the length of the string; magnitude is constant through an ideal string. Friction ($F_f$) Force opposing relative motion or tendency of motion between surfaces in contact. Static: $F_{f,s} \le \mu_s F_N$ Kinetic: $F_{f,k} = \mu_k F_N$ Air Resistance ($F_D$) Force opposing the motion of an object through air. Complex; often proportional to $v$ or $v^2$. Free-Body Diagrams (FBDs) A diagram showing all forces acting on a single object. Steps: Isolate the object of interest. Draw a coordinate system. Represent the object as a point mass. Draw and label all forces acting on the object, originating from the point mass. Do NOT include forces exerted by the object on other objects. Resolve forces into components if necessary. Applications & Problem-Solving Steps Read & Visualize: Understand the scenario, draw a sketch. Identify Objects: Determine the object(s) of interest. Draw FBDs: Create a FBD for each object. Choose Coordinate System: Align axes strategically (e.g., along motion). Apply Newton's Second Law: Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$ for each object. Solve Equations: Use algebraic techniques to find unknowns. Check Units & Reasonableness: Ensure the answer makes sense.