Unit 2 — Force and Translational Dynamics

Systems and Center of Mass

A system is an object, or a collection of objects, that you choose to analyze. If you ignore its internal structure, a system can be treated as a single object.

• Internal forces act between objects inside the system. External forces come from outside it.
• Only external forces can change the motion of the system's center of mass.
• The center of mass is the point that moves as if all the mass were concentrated there and all external forces acted on it.

x_cm = (m₁x₁ + m₂x₂ + …) / (m₁ + m₂ + …)
a_cm = F_net,external / m_total

• For a uniform, symmetric object, the center of mass is at its geometric center.
• The center of mass does not have to be located on the object (e.g. a ring or a boomerang).

Forces and Free-Body Diagrams

A force is a vector quantity describing an interaction between two objects. Its SI unit is the newton: 1 N = 1 kg·m/s². Forces can be contact (push, tension, friction, normal) or long-range (gravity).

A free-body diagram (FBD) shows all forces acting on one object:

• Draw the object as a dot or box; draw each force as an arrow starting at the object.
• Arrow length reflects magnitude; label every force with its type.
• Include only forces acting on the object, never forces it exerts on others.
• Do not draw the net force, acceleration, or "force of motion" on the FBD.

Newton's Third Law

Whenever object A exerts a force on object B, B exerts a force on A that has equal magnitude, opposite direction, and is the same type of force.

F(A on B) = − F(B on A)

• The two forces act on different objects, so they never cancel each other on a single FBD.
• Equal forces do not mean equal accelerations. a = F/m, so the smaller mass accelerates more.
• Third-law pairs exist at the same instant, and both forces are of the same kind (gravity pairs with gravity, contact with contact).

Newton's First Law

An object keeps its velocity (including a velocity of zero) unless a net external force acts on it. This tendency is called inertia, and mass is its measure.

• If ΣF = 0, the object is in translational equilibrium: it is at rest or moving at constant velocity.
• Motion at constant velocity requires no net force. A net force is needed only to change velocity.
• Equilibrium applies along each axis separately: ΣFₓ = 0 and ΣFy = 0.

Newton's Second Law

The acceleration of a system is directly proportional to the net external force and inversely proportional to its mass, and points in the direction of the net force.

a = ΣF / m        ΣFₓ = m·aₓ        ΣF_y = m·a_y

Problem-solving steps:

• Choose the system and draw its FBD.
• Pick axes (along the acceleration or along an incline) and resolve forces into components.
• Write ΣF = ma for each axis, then solve.

• Apparent weight is the normal force you feel. In an elevator accelerating upward at a: FN = m(g + a); accelerating downward: FN = m(g − a).
• Mass is a constant property; weight depends on the local gravitational field.

Gravitational Force

Every pair of masses attracts each other with a force given by Newton's law of universal gravitation:

F_g = G·m₁·m₂ / r²        G = 6.67 × 10⁻¹¹ N·m²/kg²
g = G·M / r²             (field strength at distance r from mass M)

• Gravity follows an inverse-square law: doubling r cuts the force to one quarter.
• r is measured between the centers of the masses.
• Near Earth's surface, Fg = mg with g ≈ 9.8 m/s². Gravitational mass and inertial mass are numerically equal.
• All objects in free fall have the same acceleration g, regardless of mass, because m cancels in a = F/m.

Kinetic and Static Friction

Friction is a contact force parallel to the surface that opposes relative sliding between two surfaces.

• Typically μs ≥ μk: it takes more force to start an object moving than to keep it moving.
• Friction depends on the normal force and the surfaces, and not on contact area or speed (in this model).
• The normal force is not always mg. On an incline, FN = mg cos θ; with a vertical pull or push it changes.
• Static friction can point in the direction of motion, such as a car's tires pushing the car forward.

Spring Forces

An ideal spring exerts a restoring force proportional to how far it is stretched or compressed from its equilibrium length. This is Hooke's law:

F_s = − k·x

• x is the displacement from equilibrium length; k is the spring constant in N/m (stiffer spring = larger k).
• The negative sign means the force always points toward equilibrium.
• On a graph of force vs. displacement, the slope equals k.
• For a mass hanging at rest on a spring: kx = mg, so x = mg/k.

Circular Motion

In uniform circular motion, an object moves in a circle at constant speed. Its velocity is always tangent to the circle, so its direction keeps changing. That means it is accelerating, even though its speed is constant.

a_c = v² / r            (toward the center)
ΣF_c = m·v² / r        (net force toward the center)
v = 2πr / T             (T = period)

• The centripetal force is not a new kind of force. It is the net force (or component of net force) pointing toward the center, supplied by tension, gravity, friction, a normal force, etc.
• Never draw "centripetal force" as an extra arrow on an FBD.
• If the inward force disappears, the object moves off in a straight line tangent to the circle, not outward.
• Doubling the speed requires four times the centripetal force at the same radius.