Module 2: Forces and Dynamics

This lesson aligns with the AAMC’s official MCAT content outline, specifically covering Foundational Concept 4 and Content Category 4A. Forces and dynamics — including Newton’s laws, friction, tension, and circular motion – are fundamental principles in MCAT physics. These concepts form the backbone of many passages in the Chemical and Physical Foundations of Biological Systems (C/P) section, especially when analyzing motion in biological systems. Mastery of forces not only strengthens your understanding of real-world applications but also improves your ability to reason through complex MCAT problems. You can review the official AAMC outline for this topic here.

Introduction to Forces and Dynamics

Welcome to Module 2, an essential follow-up to Kinematics. In Kinematics, we focused on describing motion, now, we dive into why objects move (or stay still). Forces and Dynamics revolve around Newton’s Laws, friction, tension, normal force, and other fundamentals that control motion.

Learning Objectives

By the end of this module, you should be able to:

• Identify and classify forces acting on an object (e.g., weight, normal, tension, friction).
• Apply Newton’s Three Laws to analyze motion or equilibrium.
• Draw and interpret free-body diagrams for one- or multi-object systems.
• Solve problems involving friction, inclines, pulleys, and circular motion.
• Recognize common mistakes (e.g., mixing up mass vs. weight) and apply tips and tricks.

Why Forces and Dynamics Matter on the MCAT

1. Frequent Appearances: MCAT passages and stand-alone questions often require analyzing whether an object accelerates or remains in equilibrium. Knowing forces is a must.
2. Real-World Ties: Many biology/physiology crossovers (e.g., blood flow pressure, tension in muscles) rely on Newton’s laws in disguised forms.
3. Foundation for Complex Topics: Circular motion (Module 3 or 4), Work-Energy (Module on Work, Energy, and Power), and even torque all build on these core principles.

Module Overview of Forces and Dynamics:

1. Newton’s Laws of Motion
2. Displacement, Velocity, and Acceleration
3. Kinematic Equations
4. Graphical Analysis of Motion
5. Projectile Motion
6. Free-Fall Motion
7. Summary & Key Takeaways

Newton’s Laws of Motion

Newton’s Laws are the backbone of classical mechanics, explaining how forces produce or prevent changes in motion.

Newton’s First Law (Law of Inertia)

Newton’s First Law states: “An object will maintain its state of rest or uniform straight-line motion unless acted upon by a net external force.”

• If nothing pushes or pulls on an object (no net force), it won’t change what it’s doing.
• A stationary object stays stationary.
• A moving object keeps moving at the same speed in the same direction.

Force is required only to change motion, not to maintain it. Without an external force, objects naturally persists in whatever motion they already have.

• Rest is not “special”.
  • Rest is just a special case of constant velocity where velocity = 0. Motion at constant speed in a straight line is just as “natural” as staying still.
• Acceleration signals a net force.
  • If you observe an object speeding up, slowing down, or changing direction, you know immediately that some net force must be acting.
• Inertia is resistance to change.
  • All objects resist changes to their motion.The more massive an object, the more it resists changes (greater inertia).

If the sum of all external forces on an object is zero:

$$
\sum F = 0
$$

Then:

$$
a = 0
$$

Meaning:

• No acceleration
• Constant velocity (could be zero or nonzero).

Examples:

• Frictionless Puck: A hockey puck on ideally frictionless ice sliders forver if no net force acts.
  • Book on a Car Seat: When the car suddenly stops, the book keeps moving forward (inertia) if not restrained.
  • Satellites in Space: With negligible external forces, a satellite continues moving at constant velocity around Earth if gravitational force is balanced or absent in the scenario considered (though typically orbits do have gravitational centripetal force, but you can imagine a deep-space scenario).

Tip: Check if sum of forces = 0 to confirm no acceleration. This also signals an equilibrium condition.

Newton’s Second Law

Newton’s Second Law states:

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

• Force causes changes in motion –force creates acceleration.
• The greater the force applied to an object, the greater its resulting acceleration
• For a given force, a more massive object will experience less acceleration.

Force and acceleration are directly proportional; mass and acceleration are inversely proportional.

• Net Force matters, not individual forces.
  •  It’s the vector sum of all external forces that determines acceleration.
• Acceleration direction matches net force direction.
  • If the net force points right, the object accelerates to the right.
• Mass is inertia in a quantitative sense.
  •  Mass measures an object’s resistance to acceleration when a force is applied

The core equation of Newton’s Second Law:

$$
\sum F = \text{Net external force (Newtons, N)}
$$

$$
m = \text{Mass (kilograms, kg)}
$$

$$
a = \text{Acceleration (meters per second squared, m/s²)}
$$

Examples:

• Pushing a Box: Applying a net force on a box will cause it to accelerate in the direction of the force.
• Heavy vs. Light Objects: Pushing a heavy crate produces less acceleration compared to pushing a small box with the same force.
• Car Acceleration: The engine force propels the car forward; the larger the force, the greater the acceleration.

Where does the unit “Newton” (N) Come From?

In SI units, the Newton (symbol: N) is the standard measure of force. One newton corresponds precisely to the net force required to accelerate a mass of 1kg by 1 m/s²:

1 N = (1kg) ((m)/s²))

Tip: When you see forces labeled as “6N,” “10N,” etc., that’s shorthand for (kg)(m/s²).

Newton’s Third Law

Newton’s Third Law states:

“For every action force, there is an equal and opposite reaction force.”

• Forces always come in pairs:
  • If object A exerts a force on object B, then object B exerts a force of equal magnitude but opposite direction on object A. These two forces act on different objects –not on the same object.
  • Action and reaction forces are always:
    • Equal in magnitude
    • Opposite in direction
    • Of the same type (e.g. both gravitational, both normal forces, etc.)

Interaction pairs are fundamental to all force interactions. Even though forces are equal and opposite, they do not cancel out because they act on different objects.

• Example mistake: Students often mistakenly add action-reaction forces together –but since they act on different objects, they must be treated separately when applying F=ma.

The equation representing Newton’s Third Law is:

$$
F_{\text{A on B}} = -F_{\text{B on A}}
$$

Where:

• F_{A on B} is the force that object A exerts on object B,
• F_{B on A} is the force that object B exerts on object A.
• The negative sign indicates that forces are in opposite directions.

Examples:

• Pusing on a Wall: You push the wall to the right with 50 N, and the wall pushes you back to the left with 50 N.
• Earth and You: You pull Earth upward with the same gravitational force that Earth pull you downward –but because Earth’s mass is enormous, its acceleration is imperceptible.
• Rocket Launch: Exhaust gases are expelled downward, which cause the rocket to experience an equal and opposite upward thrust.

Always Identify:

• What two objects are interacting
• What type of force is involved (contact force, gravitational force, etc.)
• Recognize that each object experiences its own separate force and must be analyzed individually using F=ma.

Reaction forces never cancel action forces because they act on different objects.

Common Forces

Weight (mg)

Weight is the gravitational force acting on an object’s mass near the surface of the Earth.
Although “mass” and “weight” are often used interchangeably in casual conversation, they are distinct concepts in physics

• Mass is an intrinsic property of matter — a measure of the amount of material an object contains — and remains constant regardless of location.
• Weight, on the other hand, is a force that depends both on the object’s mass and the strength of the local gravitational field.

Near the Earth’s surface, gravity exerts a nearly constant downward pull on all objects. This pull, directed toward the center of the Earth, is what we call weight.

Formula for Weight:

$$
W = mg
$$

$$
W = \text{Weight (Newtons, N)}
$$

$$
m = \text{Mass (kilograms, kg)}
$$

$$
g = \text{Gravitational acceleration (9.8 m/s² near Earth’s surface)}
$$

Important Features of Weight

Direction: Always vertically downward, toward the center of the Earth.

Magnitude: Depends on both the mass and the local gravitational field strength.

Whether you are standing on the ground, jumping into the air, riding in an elevator, or floating in orbit, the gravitational force remains directed downward.
However, the apparent forces (like normal force) may vary depending on acceleration — but weight itself stays the same unless g changes.

In Newton’s Second Law, weight is treated as one of the external forces contributing to the object’s overall net force:

$$
\sum F = ma
$$

Weight is the reason that normal forces (from surfaces) or tension forces (from ropes) often need to exist, they work to balance or counteract gravity in many everyday situations.

Mass remains constant everywhere, on Earth, the Moon, Mars, or in deep space.

Weight varies depending on the strength of the local gravitational field.

Example:

1. A 10 kg object near Earth’s surface:
   • Weight = (10)(9.8) = 98 N, directed downward
2. If you took the same 10 kg mass to the Moon (g = 1.6 m/s²), weight would be (10)(1.6) = 16 N. (Only about 1/6 of its Earth weight.)

Students sometimes say “I weight 60 kg.” In physics terms, 60 kg is mass, not weight. True weight would be: (60) (9.8) = 588 N.

Always confirm if a question is asking for mass (kg) or weight (N).
Confusing the two leads to easy mistakes, especially in dynamics and force problems.

Normal Force

The normal force is the force a surface exerts to support the weight of an object resting on it. It acts perpendicular (normal) to the surface and prevents objects from falling through it. The word “normal” in this context comes from geometry — it means “at a right angle.” Normal force is a reaction force that only exists when an object is in contact with a surface. It arises as a response to other vertical forces, typically the object’s weight.

Although the normal force often equals the weight of the object, this is not always true. The magnitude of the normal force can increase or decrease depending on other vertical forces acting on the object. For example, if an object is on an inclined plane, the normal force is less than the full weight and depends on the angle of the incline. In an accelerating elevator, the normal force may be greater or less than weight, depending on the direction of acceleration. The key idea is that the normal force adjusts to ensure the object does not move through the surface — it responds to the net vertical forces acting on the object.

Understanding the normal force is essential for solving a wide range of problems involving friction, inclined planes, and vertical acceleration. In Free Body Diagrams, the normal force is drawn perpendicular to the surface, and it plays a critical role in balancing other forces to maintain equilibrium or to help determine the net force when the object is accelerating vertically.

Formula for Normal Force (in common cases)

$$
\text{Flat surface (no vertical acceleration):} \quad N = mg
$$

$$
\text{Inclined plane:} \quad N = mg \cos(\theta)
$$

$$
\text{Elevator accelerating upward:} \quad N = mg + ma
$$

$$
\text{Elevator accelerating downward:} \quad N = mg – ma
$$

Where:

$$
N = \text{Normal Force (Newtons, N)}
$$

$$
m = \text{Mass (kg)}
$$

$$
g = \text{Gravitational acceleration (9.8 m/s²)}
$$

$$
a = \text{Acceleration of the system (m/s²)}
$$

$$
\theta = \text{Angle of incline from horizontal (degrees)}
$$

Important Features of Normal Force

• Direction: Always perpendicular to the surface.
• Magnitude: Depends on the combination of vertical forces acting on the object (not always equal to mg).

The normal force is not a fixed value, it adapts based on the net vertical situation. It may be greater than, less than, or equal to the object’s weight depending on the context (e.g., an accelerating system, a person leaning against a wall, or a surface at an angle).

Examples:

• A 10 kg object resting on a flat floor:
  • N = (10)(9.8) = 98N, upward
• Same 10 kg object on a 30 degree incline:
  • N = (10)(9.8)cos(30degrees) = 84.9 N, upward and perpendicular to incline
• Person in an elevator accelerating upward at 2 m/s²:
  • N = (60)(9.8 + 2.0) = 708 N
  • Apparent weight > true weight
• Same person accelerating downward at 2 m/s²:
  • N = (60)(9.8 – 2.0) = 468 N
  • Apparent weight < true weight

Common Mistakes to Avoid

• Don’t assume N=mg in all cases, only true when the surface is flat and there’s no vertical acceleration.
• Don’t confuse normal force with weight, they are often equal in basic problems but arise from different causes.
• On inclined planes, always remember to use the perpendicular component of weight: mgcos(θ)

Friction

Friction is a contact force that opposes the relative motion or attempted motion between two surfaces in contact. It always acts parallel to the surface and in the direction opposite to motion (or attempted motion). Friction arises from the microscopic irregularities between surfaces and the intermolecular forces between them. It plays a critical role in real-world motion, enabling walking, gripping, driving, and preventing objects from sliding unintentionally.

There are two main types of friction: static friction and kinetic friction. Static friction resists the initiation of motion. It increases as more force is applied, up to a certain maximum value. Once that threshold is exceeded and the object begins to slide, kinetic friction takes over, it remains constant and is generally less than the maximum static friction.

The frictional force depends on two main factors: the normal force (how strongly the surfaces are pressed together) and the coefficient of friction, a value that depends on the materials in contact. Importantly, friction does not depend on the surface area of contact or the speed of sliding (at least in classical models used on the MCAT).

Understanding friction is essential for analyzing motion on flat and inclined surfaces, for determining whether an object will start moving, and for calculating the net force in systems involving contact. Friction is often the difference-maker in deciding whether an object is in equilibrium or accelerating, and must always be included in Free Body Diagrams when surfaces are involved.

Key Equations for Friction

$$
0 \leq f_s \leq \mu_s N
$$

Where:

$$
f_s = \text{Actual static friction force (Newtons, N)}
$$

$$
\mu_s = \text{Coefficient of static friction (dimensionless)}
$$

$$
N = \text{Normal force (Newtons, N)}
$$

This inequality shows that the static friction force adjusts to match the applied force, up to a maximum limit:

$$
f_{s,\text{max}} = \mu_s N
$$

This equation gives the maximum value static friction can reach before the object begins to slide.

$$
f_k = \mu_k N
$$

This equation shows that kinetic friction is constant once sliding has started.

Where:

$$
f_k = \text{Kinetic friction force (Newtons, N)}
$$

$$
\mu_k = \text{Coefficient of kinetic friction (dimensionless)}
$$

$$
N = \text{Normal force (Newtons, N)}
$$

Remember, Kinetic friction does not depend on the speed of the object or how much force is applied, it remains constant as long as the object is sliding.

Important Features of Friction

• Direction: Always opposes relative motion or attempted motion.
• Depends on:
  • The normal force between the surfaces
  • The type of surfaces (via coefficient of kinetic friction)
• Does not depend on surface area or velocity (in basic MCAT-level physics)

Examples:

• Box at rest on the floor:
  • A gentle push doesn’t move it -> static friction is balancing the applied force.
  • As push increases, friction increases until it reaches its maximum limit
• Box sliding on the floor:
  • Once motion begins, kinetic friction opposes the sliding motion with a constant value.
• Inclined plane with friction:
  • Friction opposes the motion of the object sliding down the incline (acts up the slope).

Common Mistakes to Avoid

• Don’t confuse static and kinetic friction.
  • Static friction applies before sliding starts.
  • Kinetic friction applies after motion begins.
• Static friction is not always at its maximum.
  • It varies depending on the applied force – until motion begins.
• Friction is not always “Against velocity” – it’s against relative motion.
  • On a conveyor belt or moving platform, it might point in the direction of motion to keep an object from slipping.

Tension

Tension is the pulling force transmitted through a string, rope, cable, or any flexible connector when it is pulled tight by forces acting from opposite ends. It is a contact force, meaning it only exists when a rope or string is attached and under tension. Tension always acts away from the object and along the length of the rope, pulling on the object it’s attached to.

In physics problems, tension often appears in systems involving pulleys, hanging masses, or connected blocks. If a string is massless and inextensible, and the pulley is frictionless, the tension throughout the rope is the same everywhere. This simplifies problem solving, especially in classic MCAT setups like the Atwood machine (two masses on either side of a pulley) or objects connected by a taut string.

Tension is not a fundamental force, it arises as a result of Newton’s Third Law: when one object pulls on another via a rope, the second object pulls back with an equal and opposite force through the rope. In Free Body Diagrams, tension should always be shown pulling away from the object it acts on, never pushing.

Understanding tension is essential for solving systems involving ropes and pulleys, vertical motion, elevator problems, and any case where forces are transferred through a cable or string. It is often part of the net force applied to an object and works alongside gravity, normal force, or friction.

Key Principles of Tension

• Direction: Always pulls away from the object, along the rope
• Acts through: Strings, ropes, or cables (but only when taut)
  • Magnitude: Can be found using Newton’s Second Law:

$$
F_{\text{net}} = ma
$$

• Equal throughout the rope if:
  • The rope is massless
  • There is no friction
  • The rope is inextensible

Examples

1. Hanging object: A 5 kg mass suspended from a rope at rest:
   • Tension = weight = (5)(9.8) = 49 N, upward
2. Atwood machine: Two masses on either side of a pulley (ideal conditions):
   • Use Newton’s Second Law on each mass
   • Set up system of equations to solve for acceleration and tension
3. Person holding a rope connected to a block:
   • Tension depends on how hard the person pulls
   • If the block is accelerating, to solve for tension use:

$$
F_{\text{net}} = ma
$$

 Common Mistakes to Avoid

• Tension never pushes. It’s always a pulling force.
• Tension doesn’t always equal weight. Only true if the object is hanging at rest.
• Don’t confuse tension forces on different objects. Each object must be analyzed separately with its own Free Body Diagram
• Tension may differ on either side of a pulley if the pulley has mass or friction (not common on the MCAT).

Drag, Buoyant Force

When an object is placed in a fluid, it experiences an upward force known as the buoyant force. This force is caused by the fluid pressure being greater at the bottom of the object than at the top, creating a net upward push. The buoyant force is responsible for why objects float, sink, or remain neutrally buoyant in a fluid. According to Archimedes’ Principle, the buoyant force on an object is equal to the weight of the fluid displaced by that object. If the buoyant force is greater than the object’s weight, it floats; if less, it sinks. This concept will be explored in more depth in the fluid statics and dynamics module, but it’s important to recognize it here as a type of vertical force that can appear alongside weight and tension in some systems

Spring/Elastic Force

A spring force is a restoring force exerted by a spring or elastic object when it is compressed or stretched from its equilibrium (rest) position. Springs obey Hooke’s Law for small displacements, which states that the force exerted by the spring is proportional to the displacement and acts in the opposite direction. That is, the spring always tries to return to its original length by pulling or pushing against the deformation.

Spring forces are vector quantities that depend on both the amount of stretch/compression and the stiffness of the spring, characterized by the spring constant 𝑘. On the MCAT, spring problems typically involve ideal springs with no mass or damping, and assume linear behavior under Hooke’s Law. The spring force plays an important role in oscillatory motion, equilibrium analysis, and energy conservation problems.

Formula (Hooke’s Law)

$$
F_{\text{spring}} = -kx
$$

Where:

$$
F_{\text{spring}} = \text{Restoring force (N)}
$$

$$
k = \text{Spring constant (N/m)}
$$

$$
x = \text{Displacement from equilibrium (m)}
$$

The negative sign indicates the spring force always acts in the direction opposite to displacement, pulling or pushing the object back toward equilibrium.

Important features of spring force

• Direction: Opposes the direction of stretch or compression
• Only acts when the spring is deformed
• Obeys linearity only for small displacements (MCAT assumes ideal springs)
• Restoring force means the spring “wants” to return to equilibrium.

Examples:

• A compressed spring: A spring compressed by 0.1 m with k = 200 N/m
  • Force: F= (200)(0.1) = 20 N, directed outward
• Stretched spring: Spring is stretched by 0.05 m
  • Restoring force pulls back toward center
• Horizontal Spring system: Spring connected to block on frictionless surface
  • Restoring force provides acceleration if the spring is compressed or stretched

Free-Body Diagrams and Equilibriums

A Free-Body Diagram (FBD) is a visual tool used in physics to represent all the external forces acting on a single object. In any Newtonian mechanics problem, drawing an accurate FBD is often the first and most important step toward correctly analyzing the system. It allows you to isolate the object from its surroundings and focus on the individual forces that influence its motion or maintain its equilibrium.

Each force is represented as a vector arrow pointing in the direction the force acts, and labeled appropriately (e.g., weight, normal force, tension, friction, applied force). The object itself is typically drawn as a dot or box, and only the forces acting on that object are included, not forces it exerts on other objects. Free-body diagrams make it possible to apply Newton’s Second Law effectively by turning real-world motion into solvable equations.

In many MCAT problems, you will be asked whether an object is in equilibrium, meaning the net force acting on it is zero. This occurs when all the forces on the object balance out, so it does not accelerate. This could mean the object is at rest or moving at constant velocity. Free-body diagrams are essential in checking for equilibrium and determining whether there is a net force present.

Understanding how to draw and interpret FBDs, and use them to analyze horizontal and vertical force components, is critical for solving problems involving tension, friction, inclined planes, elevators, and pulleys. It also helps avoid common mistakes, such as misplacing the direction of friction or omitting a vertical support force.

• Free-Body Diagram (FBD):
  • A diagram that shows all external forces acting on a single isolated object.

How to Approach Free-Body Diagrams

Follow these steps for drawing and analyzing FBDs:

1. Isolate the object.
   • Focus on just the object in question – mentally separate it from the environment.
2. Represent it as a dot or box.
   • Use a simple point or box to represent the object in your diagram.
3. Identify all forces acting on it.
   • Ask: What forces are acting on the object? Common ones include:
     • Weight (always downward)
     • Normal force (perpendicular to surfaces)
     • Tension (along ropes/strings)
     • Friction (opposes motion or attempted motion)
     • Applied force (from pushing, pulling, etc.)
4. Draw force arrows from the center
   • Each arrow should:
     • Point in the correct direction
     • Be labeled clearly (e.g., T, N, mg, f_k)
     • Be scaled approximately to represent relative magnitudes, if possible
5. Resolve angled forces into components (if needed).
   • Break diagonal forces like gravity on an incline into horizontal and vertical parts:
     • mg sin θ (parallel to incline)
     • mg cos θ (perpendicular to incline)
6. Apply Newton’s Second Law in each direction.
   • Write equations for ∑F_x and ∑F_y and solve as needed.

Equilibrium Condition:

An object is in equilibrium if the net force in every direction is zero. This means that if there is no net force, then no net acceleration is occurring.

• Static Equilibrium:
  • The object is at rest and remains at rest.
• Dynamic Equilibrium:
  • The object is moving at a constant velocity in a straight line (zero acceleration).

Examples of FBD Applications

1. Block at rest on a Surface:
   • Forces: weight downward, normal force upward
   • If no other forces, then ∑F_y = 0, then object is in vertical equilibrium
2. Hanging mass on a string:
   • Forces: tension upward, weight downward
   • If object is stationary or moving at constant velocity, then T = mg
3. Inclined plane with friction:
   • Forces: weight (resolved into components), normal force, friction
   • Analyze components parallel and perpendicular to the incline

Common Mistakes to Avoid

• Forgetting to include all forces, such as friction or tension
• Including forces the object exerts on others, instead of those acting on it
• Incorrectly resolving weight on inclined planes – always break into mg sin (θ) and mg cos (θ)
• Assuming equilibrium without checking net force – always apply Newton’s Second Law to verify

Inclined Planes, Pulleys, and Common Systems

An inclined plane is a flat surface tilted at an angle from the horizontal. These surfaces are common in real-world physics problems, and they introduce complexity because forces no longer act purely along vertical or horizontal axes. Analyzing motion on an incline requires resolving the gravitational force into parallel and perpendicular components relative to the slope.

What happens on an Incline?

• Gravity still acts straight downward with magnitude mg
  • But for solving problems, we break it into components:
    • One that pulls the object down the slope
    • One that pushes the object into the surface

This is why we align our axes along and perpendicular to the incline – it simplifies the math and removes the need to handle angled forces directly.

Forces Commonly Acting on Inclines:

• Gravitational Force – always straight down
  • Normal Force – perpendicular to the surface (not vertical!)
  • Friction – parallel to the surface, opposing motion or attempted motion.
• Applied Forces or Tension, if present.

Equations for Inclined Planes (and how to use them)

• Parallel Component of Gravity

$$
F_{\parallel} = mg \sin(\theta)
$$

• Use this to calculate the force pulling the object down the incline
  • This is the part of gravity responsible for sliding the object.
  • Always points down the slope, regardless of whether the object moves or not.

• Perpendicular Component of Gravity

$$
F_{\perp} = mg \cos(\theta)
$$

• Use this to find how hard the object presses into the surface
  • It determines the normal force – the support force from the surface.
  • This component does not cause sliding.

• Normal Force (on incline with no vertical acceleration)

$$
N = mg \cos(\theta)
$$

• Use this to plug into friction equations or to check equilibrium
  • Normal force is not equal to mg unless the surface is flat.
  • On inclines, it’s always less than mg.

Frictional Forces

• Static Friction:

$$
f_s \leq \mu_s N
$$

• Use this to determine whether the object remains at rest.
  • f_sincreases to match the force trying to make the object move – up to a limit.
  • The object starts to move if the required f_s exceeds μ_sN

• Kinetic Friction

$$
f_k = \mu_k N
$$

• Use this once the object is sliding.
  • Constant value, opposes motion.
  • Plug directly into the net force equation:

$$
F_{\text{net}} = F_{\parallel} – f_k
$$

• Net Force Along the Incline

$$
\sum F_{\parallel} = ma
$$

• Use this to solve for acceleration or unknown forces (like applied force or tension).
  • Total force in the direction of the slope determines the object’s acceleration.
  • Include or exclude friction based on the problem details.

Common Mistakes to Avoid

• Forgetting to resolve mg into components. Always break into sin(θ) and cos(θ)
• Assuming N = mg. This is only true on flat ground.
• Misplacing the direction of friction
• Mixing up static vs. kinetic friction
• Using incorrect signs when setting up ∑F = ma

Clarifying note to “Example: Block Sliding Up a Rough Incline”:

Although the block is moving upward after being pushed, the net force and acceleration are directed down the incline because both gravity and friction oppose the motion. This negative acceleration means the block is slowing down as it moves up, and if no other forces act, it will eventually come to a stop and reverse direction.

Pulleys

A pulley is a simple machine that changes the direction of a force applied to a rope or string. In physics problems, especially those tested on the MCAT, pulleys are usually assumed to be:

• Massless: The pulley itself has no mass.
• Frictionless: No energy is lost in the axle or rope.
• Ideal Rope: The rope is massless and inextensible (doesn’t stretch).

The key takeaway is that tension is the same throughout a single, continuous rope in these idealized setups.

Atwood Machine (Classic Pulley System)

The Atwood machine is a standard 2-mass pulley system with m1​ and m2​ hanging on either side of a frictionless, massless pulley.

• If m₁ > m₂: The system accelerates such that m₁ goes does down and m₂ goes up.
• If m₂ > m₁: Reverse happens.

We apply Newton’s Second Law to each mass separately:

Mass m₁ (if moving upward):

$$
T – m_1g = m_1a
$$

Mass m₂ (if moving downward):

$$
m_2g – T = m_2a
$$

Where:

$$
T = \text{Tension (Newtons, N)}
$$

$$
m_1 = \text{Mass 1 (kilograms, kg)}
$$

$$
m_2 = \text{Mass 2 (kilograms, kg)}
$$

$$
g = 9.8 \ \text{m/s}^2
$$

$$
a = \text{Acceleration of the system (m/s}^2\text{)}
$$

We solve these two equations simultaneously to find:

• Tension, T
• Acceleration, a

How to Use these Equations:

• Draw Free Body Diagrams (FBDs) for each mass to identify forces.
• Set up Newton’s Second Law for each mass.
• Be mindful of signs:
  • If you assume m₁ accelerates upward, the tension is upward and gravity is downward
  • For m₂ moving downward, gravity acts downward, tension acts upward 
    • Add the two equation together to eliminate T and solve for a.
    • Plug a back into one equation for T.

Key Result Formula for the Atwood Machine (optional shortcut):

Acceleration of the system:

$$
a = \frac{(m_2 – m_1)g}{m_1 + m_2}
$$

Tension in the rope:

$$
T = \frac{2 m_1 m_2 g}{m_1 + m_2}
$$

(We can skip these formulas if focusing purely on deriving from Newton’s laws, but it’s useful for fast checks!)        

What does a = 2.45 m/s² mean?

The key idea: BOTH masses are part of the same rope system.

• The rope is inextensible, meaning both masses are constrained to accelerate with the same magnitude of acceleration.

So:

• M₁: accelerates upward at 2.45 m/s²
• M₂: Accelerates downward at 2.45 m/s²

The direction is opposite, but the magnitude of acceleration is the same.

What about the tension T?

In a simple Atwood machine with a massless, frictionless pulley and an ideal rope:

• The tension is uniform throughout the same rope.
• That means the same T pulls:
  • Up on m₁
  • Up on m₂

Even though m₂ is accelerating downward, the rope still pulls up with force T (it’s just that gravity is stronger for m₂).

This is exactly why it’s so powerful to draw free-body diagrams and be crystal clear about direction conventions when solving.

Elevator Problems

Elevator problems are classic Newton’s Second Law applications where an object (often a person) is inside an elevator that is either:

• At rest or moving at constant velocity
• Accelerating upward
• Accelerating downward

These problems help us understand how apparent weight changes depending on the acceleration of the elevator.

What’s Happening Physically?

• The actual weight of an object is always W = mg (gravity’s pull).
• However, the normal force (N) from the floor of the elevator is what we feel as “apparent weight”.
• If the elevator accelerates, the normal force changes even though gravity remains constant.

Cases:

Elevator at Rest or Moving at Constant Velocity

• In this case, the elevator is either stationary or moving at a constant speed (up or down).
• Even if it’s moving no acceleration means no net force
• Newton’s 2^nd Law becomes:

$$
\sum F = ma = 0
$$

• This simplifies to:

$$
N = mg
$$

• What does this mean?
  • The normal force N exactly balances your weight
  • The scale reads your true weight
  • A common mistake students make is thinking that movement affects weight – but unless there’s acceleration, your apparent weight does not change.

Elevator Accelerating Upward

• Here, the elevator speeds up as it moves up or slows down while moving down –in both cases, the acceleration is upward.
• Newton’s 2nd Law (Elevator Accelerating Upward):

$$
N – mg = ma
$$

• Solving for normal force:

$$
N = mg + ma
$$

• What does this mean?
  • The normal force must “do extra work” to overcome gravity plus accelerate you upward.
  • The scale reads higher than your true weight (you feel heavier).
  • Example: When an elevator starts rising quickly, you feel pushed into the floor.
  • Tip: The term +ma tells us the added force required to accelerate you upward.

Elevator Accelerating Downward

• Now the elevator is speeding up going down or slowing down while moving up –in both cases, the acceleration is downward.
• Newton’s 2nd Law (Elevator Accelerating Downward):

$$
N – mg = -ma
$$

• Solving for Normal Force:

$$
N = mg – ma
$$

• What does this mean?
  • The normal force is reduced because gravity is “helping” accelerate you downward.
  • The scale reads less than your true weight (you feel lighter).
  • Example: You’ve probably felt this when an elevator starts dropping quickly, a brief “light” sensation.
  • Tip: The -ma term subtracts force from what’s needed to support you.

Elevator in Free Fall (Hypothetical)

• This is an extreme case: the elevator cable snaps and the whole system falls freely under gravity.
• Acceleration:

$$
a = g
$$

• Newton’s 2nd Law:

$$
N – mg = -mg
$$

• Solving for Normal Force:

$$
N = 0
$$

• What does this mean?
  • There’s no normal force at all.
  • You and the elevator both fall at the same rate, so you’d float weightlessly.
  • This is true weightlessness (sometimes called “apparent weightlessness”).
  • Tip: This case isn’t usually tested on the MCAT, but it helps illustrate why astronauts feel weightless in orbit – they are in constant free fall around Earth.

Remember:

• The normal force N is what the scale reads and what you physically feel.
• The equation N = mg ± tells you how the apparent weight changes:
  • +ma means you feel heavier (upward acceleration).
  • -ma means you feel lighter (downward acceleration).

For a concise summary of all key equations and insights from this module, be sure to consult the Module 2 High-Yield Notes.

Summary and Key Takeaways

Forces and Dynamics form the foundation of classical mechanics and are heavily tested on the MCAT, particularly within the Chemical and Physical Foundations (C/P) section. This module explored how forces affect motion through Newton’s Laws, free-body diagrams, friction, tension, normal forces, and equilibrium conditions. Understanding how to analyze the net force acting on a system and predict the resulting acceleration or lack thereof is a critical reasoning skill on the exam.

Key Takeaways:

• Newton’s First Law (Inertia): An object at rest or in uniform motion will remain that way unless acted upon by a net external force.
• Newton’s Second Law (F = ma): The net force on an object causes acceleration proportional to its mass. This is the most tested equation in MCAT dynamics problems.
• Newton’s Third Law (Action–Reaction): Forces always occur in equal and opposite pairs, even if they act on different objects.
• Free-Body Diagrams (FBDs): Essential tools for visually mapping forces and solving net force problems — especially in inclined planes, tension systems, and friction.
• Friction: Includes static friction (prevents motion) and kinetic friction (resists motion), both dependent on the normal force and coefficient of friction.
• Equilibrium: An object is in equilibrium when the net force and net torque acting on it are zero — leading to no linear or angular acceleration.
• Common MCAT Pitfalls: Forgetting to break vectors into components, mislabeling directions (sign conventions), or failing to include all forces in FBDs.

A solid command of force analysis — both conceptually and mathematically — is vital for navigating MCAT physics passages that involve interacting systems, contact forces, inclined surfaces, or dynamic acceleration scenarios.

Master These Before Moving On:

Before progressing to the next module, make sure you’ve thoroughly mastered the following key skills and concepts — they will show up again and again on MCAT physics passages:

• Newton’s Laws of Motion: Know how to apply each law conceptually and mathematically — especially F=ma in systems with multiple forces.
• Drawing Free-Body Diagrams (FBDs): Be able to isolate a single object and correctly label all forces acting on it, including tension, normal force, weight, and friction.
• Force Components: Practice resolving forces into x- and y-components, especially on inclined planes, a favorite on the MCAT.
• Static vs. Kinetic Friction: Understand how to use fmax=μsN and fk=μkN, and when to use each.
• Equilibrium Conditions: Be confident identifying when ∑F=0 and ∑τ=0, and how to apply them in balance problems.
• Common Mistakes: Avoid errors like neglecting gravitational force components, confusing mass with weight, or misidentifying the direction of net force.
• Tip: Many MCAT dynamics questions test reasoning just as much as equations, so practice explaining why an object speeds up, slows down, or stays in equilibrium based on the force setup.