# Kinetics Force & Acceleration of a Particle

Galileo study of pendulums and falling bodies dispelled many of the early notions of Dynamics while providing insight on the effects of forces on bodies that are in motion.  This would later allow Isaac Newton with bases for his famous three laws of motion.

1. “A particle originally at rest, or moving in a straight line with a constant velocity, will remain in the is state provided the particle is not subjected to an unbalanced force.”
2. “A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force.”
3. The mutual forces of action and reaction between two particles are equal, opposite, and co-linear.”

Laws 1 and 3 are used in the development for the concepts of Statics.  While the second law of motion is the basis of the study of Dynamics.  This is due to the fact that the second law relates acceleration to force.  Hence, if an unbalanced force F is applied to a particle, the resulting acceleration a will be proportional to the mass of the particle.  This statement is the basis of Kinetics which is the study of the forces on a particle or a rigid body.

(Eq 1)  $f=ma$

m = objects mass

a = acceleration

f = force

Force in SI units is Newtons (N) and in imperials unit forces stated in pound force (lbf).

### Example

A 1000kg car is accelerating onto the highway a 3.2 $\frac{m}{s^2}$.  What force will the wheels have to exert onto the road?  Assume that the road is flat and drag related forces are negligible.

### Solution

mass = 1000kg

acceleration = 3.2 $\frac{m}{s^2}$

$f=1000(3.2)=3200 N$

There will also be cases where the forces will be represent as Cartesian Coordinates instead of as a magnitude.  In order to understand the full force acting on an object it is beneficial to convert these forces into a magnitude.

(Eq 2)  $F=\sqrt{∑ƒ_x^2+∑ƒ_y^2+∑ƒ_z^2}$

### Example

Find the magnitude for the following forces.  There is 34lbf in the x-direction, a -23lbf in the y-direction, and a 12lbf in the z-direction.

### Solution

$F=\sqrt{34^2+23^2+12^2}=42.8lbf$

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