Unit 2: Force and Translational Dynamics

Dynamics

In AP Physics 1, dynamics is the study of the causes of motion and the changes in motion. It is a cornerstone of classical mechanics because it explains not just how objects move, but why they move. Dynamics relies heavily on Newton’s Laws of Motion, which describe the relationship between forces and motion. Understanding dynamics allows us to analyze everyday phenomena such as why a car accelerates when you press the gas pedal, why objects fall when dropped, and how forces interact in systems like elevators and pulleys. These principles also form the foundation for advanced physics and engineering topics.

Newton’s Laws of Motion

First Law: The Law of Inertia

Newton’s First Law, also known as the Law of Inertia, explains that objects naturally resist changes in their motion. An object at rest will remain at rest, and an object in motion will continue in uniform motion (a straight line at constant velocity), unless acted upon by a net external force. This principle is critical in understanding why seatbelts are necessary in cars or why planets continue orbiting in space. It shows that forces are required not to keep an object moving, but to change its state of motion. Inertia depends directly on mass, meaning heavier objects are harder to start, stop, or change direction.

Key Points

• An object at rest remains at rest, and an object in motion maintains constant velocity unless acted on by a net external force. This highlights that force is only necessary to change an object’s state of motion, not to maintain it.
• Inertia is the property of an object to resist changes in motion. Larger mass means greater inertia, so heavy objects require more force to accelerate or stop compared to lighter ones.
• Real-world examples include passengers lurching forward in a car crash because their bodies resist the sudden deceleration, illustrating that motion does not change without an external force.

Gravitational vs Inertial Mass

Gravitational Mass

Gravitational mass measures how strongly an object interacts with the gravitational field. It is determined by the gravitational force experienced by an object in a gravitational field. This mass can be calculated by comparing the gravitational pull on an unknown object to that on a known standard object. The gravitational force between two bodies depends directly on their gravitational masses and inversely on the square of the distance between them, as described by Newton’s Law of Gravitation.

\[ F = G \frac{m_1 m_2}{r^2} \]

• Gravitational mass determines the magnitude of gravitational attraction between two objects. The larger the mass, the stronger the gravitational pull it exerts.
• It is measured by comparing the weight of an object with that of a known standard under the same gravitational field, ensuring accurate proportional measurement.
• This property explains why all objects accelerate at the same rate under gravity on Earth when air resistance is negligible, regardless of their mass.

Inertial Mass

Inertial mass is a measure of how much an object resists acceleration when a force is applied. It is found using the formula \( a = \frac{F}{m} \), where \(F\) is the applied force and \(a\) is the acceleration. Essentially, inertial mass tells us how much force is needed to change an object’s motion. Experiments show that inertial mass and gravitational mass are equivalent, a principle confirmed by Einstein’s theory of relativity.

\[ a = \frac{F}{m} \]

• Inertial mass represents an object’s resistance to changes in its motion. A more massive object requires a larger force to achieve the same acceleration as a smaller one.
• It is determined experimentally by applying a known force and measuring the resulting acceleration. This method shows a direct connection between force, mass, and motion.
• The equivalence of inertial and gravitational mass is why objects of different weights fall with the same acceleration in a vacuum.

Newton’s Second Law: Law of Acceleration

Newton’s Second Law explains how the motion of an object changes when a net force acts on it. It states that the net force on an object is equal to the product of the object’s mass and its acceleration. This law is typically written as \( \vec{F}_{\text{net}} = m \vec{a} \), where \( \vec{F}_{\text{net}} \) is the vector sum of all forces acting on the object, \( m \) is the object's mass, and \( \vec{a} \) is its acceleration. The direction of the acceleration is always the same as the direction of the net force. This law is foundational in physics because it connects force, mass, and motion in a single relationship.

\[ \vec{F}_{\text{net}} = m \vec{a} \]

• The net force acting on an object is the vector sum of all individual forces. If multiple forces act in different directions, they must be combined using vector addition before applying Newton’s Second Law.
• If the net force on an object is zero, its velocity remains constant (including staying at rest), reinforcing Newton’s First Law. Only a nonzero net force causes a change in velocity — that is, acceleration.
• The magnitude of acceleration is directly proportional to the magnitude of the net force and inversely proportional to the object’s mass. Doubling the net force doubles the acceleration; doubling the mass halves it.
• The law is most often used in free-body diagram problems, where we break forces into components, calculate the net force in each direction, and then apply \( \vec{F}_{\text{net}} = m \vec{a} \) to solve for unknowns like acceleration, force, or mass.

Newton’s Third Law: Law of Action-Reaction

Newton’s Third Law states, “For every action, there is an equal and opposite reaction.” This means that whenever one object exerts a force on another, the second object exerts a force of equal magnitude and opposite direction back on the first. These forces always occur in pairs, called action-reaction force pairs, and they act on different objects, never on the same one. This law explains many everyday phenomena, such as how a rocket launches into space or why you feel a pushback when you press against a wall. Importantly, while the two forces are equal and opposite, they do not cancel each other out because they act on different objects.

• For every force exerted by object A on object B, object B exerts an equal and opposite force back on object A. These paired forces are always present in interactions, regardless of the situation.
• Action-reaction pairs act on different bodies, which is why they do not cancel each other out. For example, when you push on the ground with your foot, the ground pushes you forward instead of canceling your push.
• Real-world examples include a swimmer pushing water backward to propel forward, a rocket expelling gas downward to lift upward, or a bird flapping its wings against the air to fly upward.
• This law highlights the reciprocal nature of forces in the universe. Without the reaction force, motion would not occur; for example, walking would be impossible without the ground pushing back against our feet.

Systems and Center of Mass

In dynamics, a system is a collection of objects that we choose to analyze together. We often treat a system as a single “super-object” so we can study its overall motion without worrying about internal details. The center of mass is the point that behaves as if all of the system’s mass were concentrated there. Understanding systems and their center of mass is essential because Newton’s Laws apply most clearly when forces are considered relative to the system as a whole. This concept helps explain why rockets move even though the exhaust gases push backward and why a gymnast’s motion can be analyzed by tracking her center of mass.

• A system is defined by the objects included for analysis. Internal forces (forces objects in the system exert on each other) cancel out, while only external forces change the motion of the system’s center of mass.
• The center of mass is the weighted average location of all the mass in a system. For a uniform object, it is at the geometric center, but for irregular or composite systems, it depends on the distribution of mass.
• The center of mass moves as if all of the system’s mass were concentrated at that point and all external forces acted on it. This simplifies analysis because we can apply Newton’s Second Law directly to the center of mass.
• In multi-object systems like cars, rockets, or sports players, analyzing the center of mass helps predict stability and motion. For example, lowering a vehicle’s center of mass increases stability and reduces rollover risk.

Forces and Free-Body Diagrams

A force is a push or pull acting on an object that can cause a change in its motion. To analyze forces systematically, physicists use free-body diagrams (FBDs), which are simplified sketches showing all the external forces acting on an object. Free-body diagrams are crucial in applying Newton’s Laws because they allow us to visualize and calculate the net force, which determines the object’s acceleration. Forces are always vectors, meaning they have both magnitude and direction, so careful attention must be paid to signs and components when solving problems. Properly drawing and interpreting FBDs is one of the most important problem-solving skills in AP Physics 1.

Types of Forces

• Gravitational Force (Weight): The Earth pulls objects toward its center with a force given by \( F_g = mg \). This always acts vertically downward near Earth’s surface, and its magnitude depends on the object’s mass.
• Normal Force: The supporting force exerted by a surface perpendicular to that surface. On a flat horizontal surface with no vertical acceleration, the normal force balances the gravitational force, so \( F_N = mg \).
• Frictional Force: A force that resists sliding between two surfaces. Friction can be static (preventing motion) or kinetic (resisting motion). Its maximum static value is \( F_{s, \text{max}} = \mu_s F_N \), while kinetic friction is \( F_k = \mu_k F_N \).
• Tension Force: The pulling force transmitted through a rope, cable, or string. Tension always pulls away from the object along the direction of the rope or cable.
• Spring Force: A restoring force exerted by a compressed or stretched spring, given by Hooke’s Law: \( F_s = -kx \), where \( k \) is the spring constant and \( x \) is the displacement from equilibrium.
• Applied Force: Any external push or pull directly exerted on the object. Applied forces can be in any direction and often combine with other forces to create motion.

Drawing Free-Body Diagrams

• Start by isolating the object of interest and representing it as a simple dot or box. This helps focus only on the forces acting on that object, not on its surroundings.
• Draw and label all external forces acting on the object as arrows, with their directions showing the line of action. Internal forces between parts of the same system should not be included.
• Make sure to include forces such as weight, normal, friction, tension, and any applied forces as needed. Neglecting even one can lead to incorrect net force calculations.
• Break diagonal forces into horizontal and vertical components if the problem requires solving for acceleration or tension in multiple dimensions. This allows Newton’s Second Law to be applied separately in the x- and y-directions.
• Double-check that every action has an appropriate reaction pair, even if the reaction force acts on another object outside the system being analyzed.

Spring Forces

A spring force is the force exerted by a stretched or compressed spring. This force is governed by Hooke’s Law, which states that the force needed to stretch or compress a spring is directly proportional to the displacement from its equilibrium (rest) position. Mathematically, this is written as \( F_s = -kx \), where \( F_s \) is the restoring force, \( k \) is the spring constant (a measure of stiffness), and \( x \) is the displacement from equilibrium. The negative sign indicates that the spring force always acts in the direction opposite to the displacement, pulling the object back toward the equilibrium position. This concept is essential for understanding systems involving oscillations, equilibrium, and energy storage in elastic materials.

\[ F_s = -kx \]

• Springs exert a restoring force that opposes displacement. When stretched or compressed, the spring force tries to return the spring to its natural length, always acting in the opposite direction of the applied force.
• The spring constant \( k \) measures how stiff the spring is. A higher value of \( k \) means the spring requires more force to stretch or compress it the same distance, which is important when comparing soft vs. rigid springs.
• Hooke’s Law only applies as long as the spring is not stretched beyond its elastic limit. If the spring is overstretched, it may deform permanently, and the relationship between force and displacement no longer remains linear.
• Spring forces are often involved in horizontal and vertical systems, especially in frictionless setups or oscillating systems. In such cases, they may be the only or primary force responsible for the object’s motion.
• Spring problems frequently require breaking down forces and using Newton’s Second Law. For example, in equilibrium problems, the spring force may exactly balance gravity or another force to keep an object at rest.

Kinetic and Static Friction

Friction is the resistive force that opposes motion or attempted motion between two surfaces in contact. It arises from the microscopic roughness of surfaces and the interactions between surface molecules. There are two main types: static friction, which prevents motion from starting, and kinetic friction, which resists motion once an object is already sliding. Both depend on the normal force (the support force from a surface) and the surface’s properties, measured by the coefficient of friction. Understanding friction is crucial in physics, as it plays a role in nearly every real-world system, from cars driving on roads to people walking.

Static Friction

\[ F_s \leq \mu_s F_N \]

• Static friction prevents motion from beginning. It adjusts itself to match the applied force up to a maximum limit, at which point motion begins.
• The maximum static friction is given by \( F_{s, \text{max}} = \mu_s F_N \), where \( \mu_s \) is the coefficient of static friction and \( F_N \) is the normal force.
• Static friction is often larger than kinetic friction for the same surfaces, which is why starting to push a heavy object is harder than keeping it moving once it’s in motion.
• Real-world examples include trying to push a stalled car, holding a box on a ramp without sliding, or the grip of shoes that prevents slipping before movement begins.

Kinetic Friction

\[ F_k = \mu_k F_N \]

• Kinetic friction resists motion once an object has started sliding. Unlike static friction, it has a constant value for a given pair of surfaces.
• The formula \( F_k = \mu_k F_N \) shows that kinetic friction is proportional to the normal force, with \( \mu_k \) representing the coefficient of kinetic friction.
• Because \( \mu_k \) is usually less than \( \mu_s \), less force is needed to keep an object sliding than to initially set it into motion.
• Everyday examples include a sled sliding over snow, a hockey puck gliding on ice, or a box being dragged across the floor after it starts moving.

Circular Motion

Circular motion occurs when an object moves in a circular path, requiring a constant inward (centripetal) force to keep it turning. Although the speed of the object may remain constant, its velocity is constantly changing direction, which means it is accelerating. This acceleration is called centripetal acceleration, and it always points toward the center of the circle. The net force that produces this acceleration is the centripetal force, which is not a new kind of force but the result of real forces like tension, gravity, friction, or the normal force acting inward. Understanding circular motion is essential for analyzing phenomena such as satellites orbiting Earth, cars turning on curved roads, or riders moving in loops on roller coasters.

\[ a_c = \frac{v^2}{r} \quad \text{and} \quad F_c = \frac{mv^2}{r} \]

Key Features of Circular Motion

• Centripetal Acceleration: An object moving in a circle experiences acceleration directed toward the center, even if its speed is constant. This happens because velocity is a vector, and a constant change in direction means continuous acceleration.
• Centripetal Force: The inward net force keeping an object in circular motion is called centripetal force. It can come from tension (in a string), friction (between tires and road), or gravity (for planets in orbit).
• Formula: The magnitude of centripetal acceleration is \( a_c = \frac{v^2}{r} \), and the corresponding centripetal force is \( F_c = \frac{mv^2}{r} \). Both depend on the object’s speed \( v \), its mass \( m \), and the radius of the circle \( r \).
• No Outward Force: There is no “centrifugal force” acting outward in an inertial frame. The sensation of being pushed outward is due to inertia, as the body tries to move in a straight line while the seat or road provides the centripetal force inward.
• Applications: Circular motion principles explain why highways are banked, how amusement park rides keep riders in their seats, and why satellites remain in orbit. Engineers rely on these equations to design safe and functional systems involving rotation.

Gravitational Force

The gravitational force is the attractive force that acts between any two masses in the universe. Near Earth's surface, it gives objects their weight and causes them to fall when dropped. On a larger scale, it governs planetary orbits, tides, and the structure of galaxies. There are two ways to understand gravitational force: one using Earth’s gravitational field (\( F_g = mg \)), and one using Newton’s Universal Law of Gravitation, which applies to any two masses in space. While the first is used in most AP Physics problems near Earth, the second gives a more general view of how gravity operates universally.

Gravitational Force Near Earth

\[ F_g = mg \]

• Weight is the gravitational force acting on a mass near Earth’s surface. It is always directed downward, toward the center of Earth, with a magnitude equal to mass times gravitational field strength.
• The gravitational field strength on Earth is approximately \( g = 9.8 \, \text{m/s}^2 \), though this value can vary slightly based on altitude or location (such as at the poles vs. the equator).
• Unlike mass, which is constant everywhere in the universe, weight depends on the local gravitational field. An astronaut’s mass doesn’t change on the Moon, but their weight does, due to the Moon’s smaller gravitational acceleration.
• Weight is a force and should always be included in free-body diagrams. In most problems, it acts vertically downward and is often balanced by a normal force or contributes to net force in inclined planes or elevators.

Newton’s Law of Universal Gravitation

\[ F_g = G \frac{m_1 m_2}{r^2} \]

• This law states that every mass in the universe attracts every other mass with a force proportional to the product of the two masses and inversely proportional to the square of the distance between them.
• The gravitational constant \( G \) has a value of \( 6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \), making gravitational forces very weak unless one or both objects are extremely massive (like planets or stars).
• Even though gravitational force is always attractive, it can act in different directions depending on the system. For example, satellites feel Earth's pull inward, while Earth simultaneously feels a pull from the satellite (action-reaction pair).
• Newton’s law explains the motion of celestial bodies, like why the Moon orbits Earth and Earth orbits the Sun. These systems are bound by gravity and governed by the inverse square law shown in the equation.

Example Problems for Unit 2: Forces and Translational Dynamics

Problem 1: Frictionless Inclined Plane

A 10 kg block slides down a frictionless incline angled at 30°. Find the acceleration of the block and the normal force exerted on it.

Step 1: Draw a free-body diagram.
Forces: gravity downward, normal force perpendicular to the incline.

Step 2: Break gravity into components.
Parallel: \( mg \sin \theta \)
Perpendicular: \( mg \cos \theta \)

Step 3: Apply Newton’s Second Law along the incline.
Net force = \( F_{\parallel} = mg \sin \theta \)
\( a = \frac{F_{\parallel}}{m} = g \sin \theta \)
\( a = 9.8 \sin 30° = 4.9 \, \text{m/s}^2 \)

Step 4: Solve for normal force.
\( F_N = mg \cos \theta = (10)(9.8)(\cos 30°) \approx 84.9 \, \text{N} \)

Answer: The acceleration is \( 4.9 \, \text{m/s}^2 \) and the normal force is \( 84.9 \, \text{N} \).

Problem 2: Friction on an Inclined Plane

A 5 kg block rests on a 25° incline. The coefficient of static friction is 0.40. Will the block remain at rest?

Step 1: Forces on the block.
Weight: \( mg \), Normal: \( mg \cos \theta \), Static friction: up the incline, max \( \mu_s F_N \).

Step 2: Find parallel component of gravity.
\( F_{\parallel} = mg \sin \theta = (5)(9.8)(\sin 25°) \approx 20.7 \, \text{N} \)

Step 3: Find maximum static friction.
\( F_{s, \text{max}} = \mu_s F_N = 0.40 (5)(9.8)(\cos 25°) \approx 17.8 \, \text{N} \)

Step 4: Compare forces.
\( F_{\parallel} (20.7 \, N) > F_{s, \text{max}} (17.8 \, N) \)
The component of gravity exceeds maximum friction, so the block will slide.

Answer: The block will not remain at rest — it will slide down the incline.

Problem 3: Spring Force and Equilibrium

A 2 kg mass hangs vertically from a spring with spring constant \( k = 50 \, \text{N/m} \). How much does the spring stretch at equilibrium?

Step 1: Set forces equal at equilibrium.
At equilibrium: \( F_s = F_g \)
\( kx = mg \)

Step 2: Solve for displacement.
\( x = \frac{mg}{k} = \frac{(2)(9.8)}{50} = 0.392 \, \text{m} \)

Answer: The spring stretches by \( 0.392 \, \text{m} \).

Problem 4: Circular Motion on a Flat Curve

A 1000 kg car travels around a flat curve of radius 50 m at 14 m/s. What coefficient of static friction is required to prevent it from skidding?

Step 1: Identify the centripetal force.
The centripetal force is provided by static friction: \( F_c = F_{s} \).

Step 2: Apply formulas.
\( F_c = \frac{mv^2}{r} \)
\( F_{s, \text{max}} = \mu_s F_N = \mu_s mg \)

Step 3: Solve for \( \mu_s \).
\( \mu_s = \frac{F_c}{mg} = \frac{mv^2 / r}{mg} = \frac{v^2}{rg} \)
\( \mu_s = \frac{(14)^2}{(50)(9.8)} \approx 0.40 \)

Answer: The coefficient of static friction must be at least 0.40.

Problem 5: Two-Mass System with Pulley (Tension and Acceleration)

A 3 kg block is on a horizontal, frictionless table and connected by a light string over a pulley to a 2 kg hanging block. Find the acceleration of the system and the tension in the string.

Step 1: Identify the system and draw free-body diagrams.
- Block on table (mass \( m_1 = 3 \, \text{kg} \)): horizontal tension \( T \) pulls it right.
- Hanging block (mass \( m_2 = 2 \, \text{kg} \)): weight \( m_2g \) pulls it down, tension \( T \) pulls it up.
Assume the system accelerates with acceleration \( a \), with the hanging mass moving down.

Step 2: Write Newton’s Second Law equations for each mass.
For the table block: \( T = m_1 a \)   (Equation 1)
For the hanging block: \( m_2g - T = m_2 a \)   (Equation 2)

Step 3: Solve the system of equations.
From Eq. 1: \( T = 3a \)
Substitute into Eq. 2: \( 2(9.8) - 3a = 2a \) ⇒ \( 19.6 = 5a \) ⇒ \( a = 3.92 \, \text{m/s}^2 \)

Step 4: Find tension.
Use Eq. 1: \( T = 3a = 3(3.92) = 11.76 \, \text{N} \)

Answer: The acceleration of the system is \( 3.92 \, \text{m/s}^2 \) and the tension in the string is \( 11.76 \, \text{N} \).

Common Misconceptions in Unit 2

Forces and Motion

• Misconception: A force is always required to keep an object moving.
  Correction: According to Newton’s First Law, an object in motion will continue moving at a constant velocity unless acted on by a net external force. A force is needed only to change velocity (accelerate), not to maintain motion.
• Misconception: Action-reaction forces cancel each other out.
  Correction: Newton’s Third Law pairs act on different objects, not the same one. For example, when you push on the wall, the wall pushes back on you, but these forces don’t cancel because they act on separate bodies.

Friction

• Misconception: Static friction always equals \( \mu_s F_N \).
  Correction: Static friction adjusts up to a maximum of \( \mu_s F_N \). If the applied force is smaller, friction matches it exactly to prevent motion; only at the maximum does sliding begin.
• Misconception: Kinetic friction increases with speed.
  Correction: In introductory physics, kinetic friction is modeled as constant for a given pair of surfaces, independent of sliding speed. Real-world effects like heat and deformation are ignored at this level.

Inclined Planes

• Misconception: The normal force always equals \( mg \).
  Correction: On an incline, the normal force is reduced to \( mg \cos \theta \). This reduced normal force also lowers the maximum static friction available.
• Misconception: The acceleration down a ramp depends on mass.
  Correction: Without friction, acceleration depends only on the angle: \( a = g \sin \theta \). Mass cancels out, so heavy and light objects slide down identical ramps with the same acceleration.

Circular Motion

• Misconception: A centrifugal force pushes objects outward in a turn.
  Correction: In reality, there is no outward force in an inertial frame. The sensation of being pushed outward is due to inertia — your body wants to move straight while the seat or road provides the inward (centripetal) force.
• Misconception: Centripetal force is a new type of force.
  Correction: Centripetal force is not a new force but the net inward force from existing forces like tension, gravity, or friction, depending on the situation.