Two Fundamental Problems of Particle Dynamics

Particle dynamics has two fundamental problem types. Both use the equation $m\vec a=\sum\vec F$, but differ in which quantities are known and which must be determined.

First problem of dynamics

Given the particle's motion, its mass, and some of the forces, determine the unknown forces. Starting from $\vec r(t)$, find velocity and acceleration and then use the equations of dynamics to determine the required forces or reactions.

Second problem of dynamics

Given the forces, mass, and initial conditions, determine the motion. Set up the differential equations, integrate them, and evaluate the integration constants from the initial conditions.

First problem: procedure

1. write the prescribed motion;
2. differentiate the coordinates twice to obtain acceleration;
3. draw the free-body diagram;
4. write the components of $m\vec a=\sum\vec F$;
5. solve for the unknown forces.

Second problem: procedure

1. isolate the particle and identify the forces;
2. choose coordinates;
3. form the differential equations of motion;
4. integrate them;
5. apply the initial conditions;
6. check the resulting motion and units.

Example of the first problem

If motion along an axis is prescribed by $x=2t^2$ m for a 3 kg particle, then $a_x=4$ m/s² and the required resultant force is $F_x=ma_x=12$ N.

Example of the second problem

If a constant force $F$ acts along an axis on a particle of mass $m$, then $\ddot x=F/m$. With $x(0)=x_0$ and $\dot x(0)=v_0$, integration gives $x=x_0+v_0t+Ft^2/(2m)$.

Common mistakes

• not distinguishing what is prescribed in the two problem types;
• omitting initial conditions in the second problem;
• trying to determine force from velocity instead of acceleration;
• omitting unknown constraint reactions from the equations of motion.