Particle Motion Under Typical Forces

Dynamics problems repeatedly use several standard force models. Writing these forces correctly and choosing consistent directions makes the differential equation of motion much easier to formulate.

Gravity

Near Earth's surface, gravity is commonly treated as constant: $\vec F_g=m\vec g$. If air resistance is neglected, vertical motion has constant acceleration $g$ directed downward.

Elastic force

For a linear spring within Hooke's-law behavior, the restoring force is proportional to deformation and opposite to it:

$$F_s=-kx.$$

With no other variable forces, $m\ddot x+kx=0$ describes free harmonic oscillation.

Dry sliding friction

In the simple Coulomb model, the sliding-friction magnitude is $F_f=\mu N$, and its direction opposes relative sliding velocity. The sign of its component must be chosen from the actual direction of motion.

Resistance of a medium

At relatively low speeds, a linear model $\vec R=-c\vec v$ is often used. In other regimes, a quadratic model with resistance magnitude proportional to $v^2$ may be appropriate. The problem statement must specify or justify the model.

Motion on an inclined plane

With an axis along a plane inclined by $\alpha$ to the horizontal, the gravity component along the plane has magnitude $mg\sin\alpha$, while the normal component is $mg\cos\alpha$. If no other normal forces act, $N=mg\cos\alpha$.

Example

A body slides down an incline with friction coefficient $\mu$. Taking positive direction down the plane gives $ma=mg\sin\alpha-\mu mg\cos\alpha$, hence $a=g(\sin\alpha-\mu\cos\alpha)$.

Common mistakes

• automatically directing friction opposite to the coordinate axis rather than opposite to sliding;
• writing spring force without its restoring direction;
• assuming $N=mg$ for every geometry;
• mixing linear and quadratic resistance models.