Force and Translational Dynamics

Notes

Systems and center of mass

A system is the object or group of objects you choose to analyze. The center of mass moves as if all the system's mass were there and all external forces acted on it.

Newton's laws

1. First law (inertia): an object stays at rest or at constant velocity unless a net external force acts on it. 2. Second law: $\vec F_{net} = m\vec a$. 3. Third law: if A pushes on B, B pushes on A with an equal and opposite force. These forces act on different objects, so they never cancel each other.

Free-body diagrams

Draw each force as an arrow from the object's center. Common forces:

• Weight (gravity): $F_g = mg$, down.
• Normal force: perpendicular to the surface. It isn't always $mg$ (on a ramp, $F_N = mg\cos\theta$; in an accelerating elevator it changes).
• Tension: pulls along a rope.
• Friction: parallel to the surface, opposing sliding. Kinetic: $f_k = \mu_k F_N$. Static: $f_s \le \mu_s F_N$.
• Spring force: $F_s = -kx$ (Hooke's law).

Gravitation

$\displaystyle F_g = \frac{Gm_1m_2}{r^2}$. Doubling distance makes the force $\displaystyle \frac{1}{4}$ as large. Near Earth, $\displaystyle g = \frac{GM}{R^2} \approx 9.8$ m/s².

Circular motion

• An object moving in a circle at constant speed accelerates toward the center: $\displaystyle a_c = \frac{v^2}{r}$.
• The net force toward the center is $\displaystyle F_c = \frac{mv^2}{r}$. It's provided by real forces like tension, friction or gravity, not a separate "centripetal force."
• Period $\displaystyle T = \frac{2\pi r}{v}$.