Fundamentals of Physics/The Laws of Motion

In this chapter, we use Newton's Laws to see that accelerations actually come from forces applied to an object. In the chapters on one and two dimensional motion, $a$ was simple a "given" quantity. It simply existed in the equations of motion and you were allowed to give it any value. In this chapter, we will see that accelerations come from forces. Forces are pushes or pulls on objects that you witness everyday (push a door to open it, pull on your book to lift it, push a cell phone button to click it). With the exception of gravity and electromagnetic fields, forces are always "contact forces" meaning a force must actually touch an object to exert its influence on it. Forces also require an agent, meaning that you should always be able to identify what (the agent) is producing the force. Forces are also vectors, meaning their strength (push or pull) can be in any direction. The SI unit of force is aptly named the Newton, and is symbolized by N.

Newton's Laws

You probably know that Newton's Law says $F=ma$, but this is not always the most useful equation to try and use in a physics course. $a={\frac {F}{m}}$ is better, but still isn't quite right. It is more correct to say that $a={\frac {\Sigma F}{m}}$, which still isn't fully correct. The best version is ${\vec {a}}={\frac {\Sigma {\vec {F}}}{m}}$, stressing the vector property of forces. Be sure you fully understand what this last version means and how to use it. ${\vec {a}}$ is the acceleration, $m$ is the object's mass and $\Sigma {\vec {F}}$ is the sum of all forces acting on the object.

Types of forces

We will only be concerned with five forces: weight, tension, normal, friction, and drag.

Weight

Weight is $w=mg$, where $m$ is an object's mass and $g$ is the Earth's acceleration of gravity of $9.8$ m/s². The weight $W$ of a body is equal to the magnitude $F_{g}$ of the gravitational force on the body. Do not confuse mass and weight; they are not the same thing and be sure you know the difference between them. Mass is also known as object's inertia, or resistance to want to change its current state of motion. A person who has mass 80 kg, has mass 80 kg on Earth, and on Jupiter. However, the person's weight on Jupiter is substantially greater due to the magnitude of the gravitational force on Jupiter.

Tension

If a cord is attached to an object, and then the cord is drawn taut, the force experienced by the object is called tension. The tension vector has the same direction as motion vector, meaning that tension is only experienced when an object is pulled, never when it is pushed.

Normal Force

Normal is the force an object feels when it is sitting on a surface, is always perpendicular to the surface, and is not always equal to $mg$.

Friction

Friction is a force that always opposes all motion; it always acts in a direction exactly opposite to that in which an object is moving (or trying to move), and typically comes when the surface rubs or drags against another as it moves. It is defined as ${\vec {f}}=\mu {\vec {N}}$ where $\mu$ is the coefficient of friction, and ${\vec {N}}$ is the normal force acting on the object. There are two types of friction, static and kinetic, also known as sliding friction.

Static friction is equal to the amount of initial force required to move an object, and kinetic friction represents the minimum amount of force required to sustain motion of an object. Because of this, static friction is always greater than or equal to kinetic friction.

Drag

Drag is the resistance that a fluid offers to an object moving in it. Fluid in this case also applies to any substance that acts like a fluid, for instance air. It is also sometimes called air resistance or fluid resistance.

Drag forces depend on velocity, they always acts in a direction opposite to that of oncoming flow velocity (e.g. as a plane is passing trough air, the air will generate a drag force). This relationship is documented in the equation $D-F_{g}=ma$, because the vectors representing drag and motion are in opposite directions.

The drag force on an object is proportional to the density of the fluid and to the square of the relative speed between the object and the fluid, and so drag is related by the equation $D={\frac {1}{2}}CpAv^{2}$, where $C$ is the drag coefficient, $A$ is the effective surface area that encounters the resistance, $p$ is the fluid density, and $v$ is the object's speed.

The key point

The crux of this entire chapter is $\Sigma {\vec {F}}$, because it requires three hard steps. The first, which most students have great difficulty with, is to identify all forces acting on an object. The second is to correctly draw these forces, each pointing in the proper direction, as they act on the object (even more difficult for most students). The third is to realize that $\Sigma {\vec {F}}$ which is only usable when you break it up into component form, or $\Sigma F_{x}$ and $\Sigma F_{y}$. Your working equations for this week are then $a_{x}=\Sigma F_{x}/m$ and $a_{y}=\Sigma F_{y}/m$.

Connections with kinematics (an object's observed motion)

The connection points with the chapters on one and two dimensional motion are that these accelerations, which come from forces, are the same a's that go into the equations of motion for $x$ and $y$. Thus, $x=x_{0}+v_{0}\Delta t+\Sigma F_{x}\Delta t^{2}/(2m)$ and $v_{x}=v_{0x}+\Sigma F_{x}\Delta t/m$. Be sure these make sense to you and do not casually read over the $\Sigma F_{x}$ and $\Sigma F_{y}$. Know what they mean: Using Newton's law is a way of finding the acceleration on an object, and this really means that all forces acting on an object need to be broken up into their $x$ and $y$ components, properly signed (+ or -), then added together along a given axis to then find a_x and a_y.