Forces and Acceleration

A force is any interaction that, when unopposed, changes the motion of an object. Every change in velocity — speeding up, slowing down, or changing direction — is caused by a push or a pull, which is exactly what a force is.

Key Takeaways

• A force is any push or pull that can change an object's motion; measured in Newtons (1 N = 1 kg·m/s²).

• Forces are either contact (friction, tension, normal force) or non-contact (gravity, magnetic, electrostatic).

• Gravity follows F = G(m₁m₂)/r²; weight is the specific case of gravity acting on an object's mass, W = mg.

• Static friction (0 ≤ fs ≤ μsN) resists the start of motion; kinetic friction (fk = μkN) resists motion already in progress, and is usually smaller than maximum static friction.

• Mass is constant and scalar; weight is variable (with gravitational field strength) and vector.

• Center of mass is the single point where an object's mass can be treated as concentrated for motion analysis: x_cm = (Σmᵢxᵢ)/(Σmᵢ).

What Is a Force?

Force (F) is a vector quantity: it has both magnitude and direction, and it's responsible for causing accelerations. The unit for force is the Newton (N), defined as:

1 N = 1 kg·m/s²

Contact Forces vs. Non-Contact Forces

Gravity is the most familiar non-contact force, and friction is a classic example of a contact force — both are explored in more depth below.

Gravity and Weight

Gravity is one of the fundamental forces of nature: an attractive force that acts between all objects with mass. Anything with matter exerts a gravitational pull on everything else with matter. The magnitude of this force depends on two factors — the masses of the objects involved and the distance between them — expressed by Newton's Law of Universal Gravitation:

F = G(m₁m₂) / r²

where G is the gravitational constant (≈ 6.674 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are the masses of the two objects, and r is the distance between their centers. The larger the masses, the stronger the gravitational pull; the farther apart they are, the weaker it becomes.

The Earth's large mass exerts a significant gravitational force on nearby objects — the force we experience as weight. Weight is simply the force of gravity acting on an object's mass:

W = mg

where g is the acceleration due to gravity, approximately 9.8 m/s² near the Earth's surface. Gravity is what keeps planets in orbit, governs the motion of celestial bodies, and gives objects weight here on Earth.

Friction

Friction is a force that opposes the motion of objects. It acts parallel to the surface of contact and works against the direction of movement. There are two main types.

Static Friction

Static friction (fs) occurs between a stationary object and the surface it rests on. It prevents the object from moving and must be overcome by an external force to initiate motion. It can vary up to a maximum value:

0 ≤ fs ≤ μsN

where μs is the coefficient of static friction and N is the normal force.

Kinetic Friction

Kinetic friction (fk) comes into play once an object is already moving, acting between the moving object and the surface it slides over:

fk = μkN

where μk is the coefficient of kinetic friction. Kinetic friction is usually lower than the maximum static friction — which is why less force is needed to keep an object moving than to start it moving in the first place.

MCAT Callout — Three General Principles of Friction: (1) Proportional to normal force — frictional force increases as the normal force increases (this is why both friction formulas include N). (2) Independent of contact area — friction doesn't change based on how much surface area is in contact, as long as the normal force and materials stay the same. (3) Independent of relative velocity (for kinetic friction) — once an object is moving, kinetic friction stays roughly constant regardless of how fast it's moving.

Mass vs. Weight

Mass is a measure of the amount of matter in an object, reflecting its inertia — its resistance to changes in motion. Weight, by contrast, is the force exerted on an object due to gravity, always pointing toward the center of the Earth (or whatever body is exerting the gravitational pull). Mass stays constant no matter where an object is; weight changes depending on the local gravitational field.

Center of Mass

Center of mass is the point in an object where all its mass can be considered concentrated for the purpose of analyzing linear motion. Rather than tracking the forces on every individual particle in an object or system, it's often far simpler to analyze the forces acting on its center of mass.

For a system of particles along the x-axis, the center of mass is calculated as:

x_cm = (Σmᵢxᵢ) / (Σmᵢ)

— the sum of each particle's mass multiplied by its position, divided by the total mass. Center of mass is especially important when analyzing rotational motion and when evaluating the stability of objects.

Common MCAT Mistakes

• Mixing up mass and weight. Mass (kg) is a scalar that never changes; weight (N) is a vector force that depends on local gravitational field strength. A 10 kg object has the same mass on the Moon as on Earth, but far less weight.

• Assuming kinetic friction is always used once something is moving. Static friction governs the threshold to start motion (up to μsN); kinetic friction only applies once the object is already sliding. Test questions often ask for the minimum force to initiate motion, which uses the static — not kinetic — coefficient.

• Thinking a larger contact area means more friction. Both friction formulas depend only on the normal force and the relevant coefficient of friction — not on how much surface area is touching.

• Reaching for F = G(m₁m₂)/r² when W = mg is all that's needed. Newton's Law of Universal Gravitation is for the general case between two masses; near Earth's surface, weight simplifies to W = mg using g ≈ 9.8 m/s².

MCAT-Style Concept Check

Question: A 10 kg box rests on a horizontal floor. The coefficient of static friction between the box and floor is 0.4, and the coefficient of kinetic friction is 0.3. Using g = 10 m/s², what is the minimum horizontal force required to start the box moving from rest?

• A) 30 N

• B) 40 N

• C) 100 N

• D) 400 N

Answer: B

Explanation: The normal force is N = mg = (10 kg)(10 m/s²) = 100 N. To start the box moving from rest, the applied force must overcome the maximum static friction: fs,max = μsN = (0.4)(100 N) = 40 N. The kinetic friction coefficient (0.3) only becomes relevant once the box is already sliding — it doesn't determine the force needed to initiate motion.

FAQ

What's the actual difference between mass and weight?

Mass is a scalar quantity measuring the amount of matter in an object (in kilograms) and never changes based on location. Weight is a vector quantity — the force of gravity acting on that mass (in Newtons, Fg = mg) — and it changes depending on the strength of the local gravitational field.

What's the difference between static and kinetic friction?

Static friction acts on an object at rest and resists the start of motion, varying up to a maximum of μsN. Kinetic friction acts on an object already in motion and is calculated as fk = μkN. Kinetic friction is usually smaller than the maximum static friction, which is why it takes more force to start an object moving than to keep it moving.

Does the size of the contact area affect friction?

No. Friction depends only on the normal force and the coefficient of friction between the two surfaces — not on how much surface area is in contact, as long as the materials and normal force stay the same.

Why does center of mass matter?

Center of mass is the single point where an object's entire mass can be treated as concentrated for analyzing motion. It simplifies force analysis by replacing a complex system of particles with one representative point, and it's especially useful when evaluating rotational motion and object stability.