Forces and Translational Dynamics: Study Guide for PHYS 210 Exam 2

Forces and Translational Dynamics

Newton’s Laws of Motion

Newton’s Laws of Motion are fundamental principles that describe the relationship between the motion of an object and the forces acting upon it.

• Newton’s 1st 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.

• Newton’s 2nd Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The law is mathematically expressed as: $F_{\text{net}} = m a$

• Newton’s 3rd Law (Law of Action and Reaction): For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A.

Newton’s 2nd Law with Multiple Forces

When multiple forces act on an object, the net force is the vector sum of all individual forces. The acceleration is determined by this net force:

• Net Force: $\sum \vec{F} = m \vec{a}$

• Each force must be considered with its direction (vector addition).

• Free-body diagrams are essential tools for visualizing and summing forces.

Forces in Two Dimensions and Free-Body Diagrams

Forces often act in more than one dimension. Vector mathematics is used to resolve forces into components and analyze their effects.

• Vector Components: Any force can be broken into x and y components using trigonometry.

• Free-Body Diagram (FBD): A diagram showing all external forces acting on an object. Each force is represented as an arrow pointing in the direction of the force.

• Sum forces in each direction separately to solve for unknowns.

Normal Force

The normal force is the perpendicular contact force exerted by a surface on an object resting on it.

• On a horizontal surface: $F_N = mg$ (if no other vertical forces are present)

• On an inclined plane: $F_N = mg \cos \theta$

• The normal force adjusts to balance other vertical forces, preventing penetration of surfaces.

Friction Force

Friction is a resistive force that opposes the relative motion of two surfaces in contact.

• Static Friction ($f_s$): Prevents motion up to a maximum value: $f_s \leq \mu_s F_N$ where $\mu_s$ is the coefficient of static friction.

• Kinetic Friction ($f_k$): Opposes motion once sliding begins: $f_k = \mu_k F_N$ where $\mu_k$ is the coefficient of kinetic friction.

• Typically, $\mu_s > \mu_k$.

Spring Force (Hooke’s Law)

Springs exert a restoring force proportional to their displacement from equilibrium, described by Hooke’s Law:

• $F_s = -k x$ where $k$ is the spring constant and $x$ is the displacement from equilibrium.

• The negative sign indicates the force is directed opposite to the displacement.

Resistive Forces: Drag

Drag is a resistive force experienced by objects moving through fluids (liquids or gases).

• For low speeds: $F_{\text{drag}} = -bv$ (proportional to velocity)

• For higher speeds: $F_{\text{drag}} = -cv^2$ (proportional to velocity squared)

• Drag always acts opposite to the direction of motion.

Newton’s Law of Gravitation

Newton’s Law of Universal Gravitation describes the attractive force between two masses:

• $F_g = G \frac{m_1 m_2}{r^2}$ where $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses, and $r$ is the distance between their centers.

• This force acts along the line joining the centers of the two masses.

Circular Motion

When an object moves in a circle at constant speed, it experiences a net force directed toward the center of the circle (centripetal force).

• Centripetal Force: $F_c = m \frac{v^2}{r}$ where $m$ is mass, $v$ is speed, and $r$ is the radius of the circle.

• This force can be provided by tension, gravity, friction, or other forces depending on the context.

Orbital Motion

Orbital motion is a special case of circular motion where gravity provides the centripetal force, such as planets orbiting the sun or satellites orbiting Earth.

• Set gravitational force equal to centripetal force for circular orbits: $G \frac{M m}{r^2} = m \frac{v^2}{r}$

• Solve for orbital speed: $v = \sqrt{\frac{G M}{r}}$

Example: Free-Body Diagram for a Block on an Inclined Plane

• Forces acting: gravity (downward), normal force (perpendicular to surface), friction (opposing motion).

• Resolve gravity into components parallel and perpendicular to the incline.

• Apply Newton’s 2nd Law in each direction to solve for acceleration or unknown forces.

Additional info: Students should be comfortable with vector addition, resolving forces, and applying Newton’s Laws to a variety of physical situations, including those involving friction, springs, drag, and circular motion.