Force and Translational Dynamics

A system is whatever object or group of objects you choose to study. When its inside details don't matter, you can model it as a single object located at its center of mass. For separate particles, $x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$. For a solid object you integrate, $x_{cm} = \frac{\int x\,dm}{\int dm}$, using a linear mass density $\lambda = \frac{dm}{dx}$ that may change along the object.

A force is a push or pull that comes from an interaction between two objects, so no object can exert a net force on itself. Contact forces like the normal force, tension and friction come from electric forces between atoms. A free-body diagram shows each force on one object or system as its own arrow starting from a dot. On the AP exam you draw whole forces, never components. Picking one axis along the acceleration (such as along an incline) makes the equations simpler.

When object A exerts a force on object B, B exerts a force of the same size in the opposite direction on A, so these paired forces act on different objects and never cancel on one free-body diagram. Forces between parts inside a system can't change the motion of its center of mass. An ideal string is massless and doesn't stretch, so its tension is the same everywhere along it, and an ideal pulley is massless and turns without friction.

The net force on a system is the vector sum of every force on it. Newton's first law says that if the net force is zero, the system's velocity stays constant, whether it is sitting still or moving; this is called translational equilibrium. Forces can be balanced in one direction and unbalanced in another, and an inertial reference frame is one where the first law holds.

Newton's second law, $\vec{a}_{cm} = \frac{\sum\vec{F}}{m_{sys}}$, says a system's center of mass accelerates in the direction of the net external force, with a size proportional to that force and inversely proportional to the mass. Only a nonzero net external force can change the velocity of the center of mass. For connected objects, like blocks joined by a string over a pulley, you write the second law for each object (or the whole system) and solve the equations together.

Any two masses attract each other with $F_g = \frac{Gm_1m_2}{r^2}$, along the line between their centers. The gravitational field, $g = \frac{GM}{r^2}$, is that force per kilogram; near Earth's surface it's nearly constant, so your weight is mg. Your apparent weight is the normal force on you, which differs from mg when you accelerate up or down. Inside a uniform sphere, only the mass closer to the center than you pulls on you, so the force grows in proportion to your distance from the center.

Kinetic friction acts when two surfaces slide past each other, opposes the sliding, and has size $F_{f,k} = \mu_kF_N$. Static friction acts when surfaces don't slide; it takes whatever size and direction is needed to stop slipping, up to a maximum of $\mu_sF_N$. The coefficients depend on the materials, not the contact area, and $\mu_s$ is usually bigger than $\mu_k$.

An ideal spring is massless and exerts a force proportional to how far it is stretched or compressed from its relaxed length, $\vec{F}_s = -k\Delta\vec{x}$, always back toward equilibrium. Springs in parallel add, $k_{eq} = k_1 + k_2$, while springs in series combine as $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$, which is smaller than any single spring's constant.

A resistive force, like air drag, points opposite the velocity and depends on speed. In this course it's often $\vec{F}_r = -k\vec{v}$. Putting it into Newton's second law gives a differential equation you solve by separating variables, and the answers contain exponentials like $e^{-kt/m}$ that level off over time. When drag grows to balance a constant force such as gravity, the net force is zero and the object moves at its terminal velocity (for a falling object, $v_T = \frac{mg}{k}$).

An object moving in a circle has a centripetal acceleration $a_c = \frac{v^2}{r}$ toward the center. Real forces, or components of them, supply it: gravity, tension, the normal force or friction, as on a banked curve or a conical pendulum. If the object's speed is changing, it also has a tangential acceleration, and its total acceleration is the vector sum of the two. In a circular orbit, gravity alone provides the centripetal force, which leads to Kepler's third law, $T^2 = \frac{4\pi^2}{GM}r^3$.