Dynamics Summary

• Here are Newton's Laws, as you must learn them:
  1. Law of Inertia: Objects at rest or in uniform motion (at a constant velocity) remain so unless acted upon by an unbalanced (net) force. We can write this algebraically as
     

     $$\sum_i \vec{F_i} = 0 = m \vec{a} = \frac{d \vec{v}}{dt}$$ (2.1)

     
     

  2. Law of Dynamics: The net force applied to an object is directly proportional to its acceleration. The constant of proportionality is called the mass of the object. We write this algebraically as:
     

     $$\vec{F} = \sum_i \vec{F_i} = m \vec{a}$$ (2.2)

     
     

  3. Law of Reaction: If object A exerts a force $\vec{F}_{AB}$ on object B, then object B exerts an equal and opposite reaction force of $\vec{F}_{BA} = -\vec{F}_{AB}$ on object A. We write this algebraically as:
     

     $$\vec{F}_{ij} = - \vec{F}_{ji} \quad \quad$$ (2.3)

     $${\rm (or)} \quad \sum_{i,j} \vec{F}_{ij} = 0$$ (2.4)

     
     
     (the latter form means that the sum of all internal forces cancel).

• Forces of Nature (weakest to strongest):
  1. Gravity
  2. Weak Nuclear
  3. Electromagnetic
  4. Strong Nuclear

• Forces to know for immediate use in this chapter:
  1. Gravity (near the surface of the earth):
     

     $$\vec{F} = -m g \hat{y}$$ (2.5)

     
     
     (direction down).
  2. Spring (Hooke's Law) in one dimension:
     

     $$F_x = -k\Delta x$$ (2.6)

     
     
     (directed back to equilibrium point of unstretched spring).
  3. Normal force, points perpendicular and away from solid surface, magnitude sufficient to oppose the force of contact.
  4. Static Friction
     

     $$f_s = \mu_s N$$ (2.7)

     
     
     (directed opposite towards net force parallel to surface to contact).
  5. Kinetic Friction
     

     $$f_k = \mu_k N$$ (2.8)

     
     
     (opposite to direction of relative sliding motion of surfaces and parallel to surface of contact).
  6. Drag Forces
     

     $$F_d = -b v^n$$ (2.9)

     
     
     (directed opposite to relative velocity of motion through fluid, $n$ usually between 1 (low velocity) and 2 (high velocity).

Dynamical Recipe

To solve any of the multidimensional trajectory problems you will encounter in this course, you must therefore:

1. Draw a good picture (free body diagram).
2. Decompose the 2 or 3 dimensional equations of motion into a set of independent 1 dimensional equations of motion for each of the orthogonal coordinates by choosing a suitable coordinate system (which may not be cartesian, for some problems) and using trig/geometry. Note that a ``coordinate'' here may even wrap around a corner following a string, for example.
3. Solve the independent 1 dimensional systems for each of the independent orthogonal coordinates chosen. In many problems a constraint will eliminate one or more degrees of freedom from consideration.
4. Reconstruct the multidimensional trajectory by adding the vectors components thus obtained back up (for a common independent variable, time).
5. Answer algebraically any questions requested concerning the resultant trajectory.