chapter5

__Chapter 5 Applications of Newton's Laws__ In Chapter 4, Newton's Laws were introduced and explained. In this chapter we will further expand on Newton's Laws and apply them to situations with friction and drag forces and to objects on a curved path.

**__Friction__** Friction is a phenomenon that arises from the attraction of molecules between two surfaces that are close in contact. This traction that is caused is vital to our world. It allows us to walk or drive, it lets us change direction, and it even allows things to be held together by a screw or a nail. Despite the ridiculous amount of cases where friction helps us, there are also cases where its resistence can be annoying and even hamper what we are trying to do. To combat these problems, we often use lubricants or simply just apply more force. There are three types of friction that we are going to explore and apply Newton's Laws to. These three are as follows:

__**Static Friction**__ Static friction comes into effect when an object is at rest and someone or something tries to move it. When a horizontal force is exerted on the object, the static friction will oppose this force and balance the forces. The static friction force can vary from 0 (when there is no external horizontal net force on the object) to f s,max, depending on how great the horizontal force is. f s,max is given by this equation:

f s, max = µ s F N

µ s is the coefficient of static friction and depends on the nature of the surfaces in contact. The rougher the surfaces, the higher the mu constant. Static friction is special in that it only balances the horizontal force. There fore it can vary from zero to the max static equation listed above. This is the case because otherwise the ground would arbitrarily propel things like a treadmill, which does not make sense. As a result, we can say:

f s ≤ µ s F N

Kinetic Friction comes into play when the horizontal force on the object is greater than the max static friction force. As a result, the object moves and the friction force is now kinetic since it is doing negative work on a moving object. Kinetic friction always opposes motion. In order to determine the speed of an object that is affected by kinetic friction, we must use Newton's Second Law, F = ma. In order for the acceleration to be zero and thus the velocity be constant, the horizontal force has to equal to the kinetic friction force. This is the case because it would make the net force 0, and when solving 0 = ma, either the mass or the acceleration must be 0 but obviously the mass is not 0. When the horizontal force is greater than the kinetic friction force, the net force is equal to the horizontal force - the friction force. Then, we can use Newton's second law again to find the acceleration. The force of kinetic friction can be given by this equation:
 * __Kinetic Friction__**

f k = µ k F N

The coefficient of kinetic friction is found to be less than the coefficient of static friction on the same surface. This makes sense because think of sliding a big box on the ground. It usually takes quite a bit of force to get it going, but once it is moving, it tends to slide much more easily.

__**Rolling Friction**__ Rolling friction affects wheels that are rolling on a surface. It is this type of friction which creates the traction that allows a car to drive forward or backward or any other type of wheel roll. Without rolling friction, wheels would just spin about in one fixed spot (sort of like when your car is stuck in mud or snow). The equation for rolling friction is as follows:

f r = µ r F N

As this Ferarri is rounding the curve during a race, it is the static frictional force exerted by the road on the tires that prevents it from sliding out radially and the force of rolling friction that allows it to drive at blistering speeds. This is also an example of circular motion, which we will get into in the next section.

__**Motion Along a Curved Path**__ In Chapter 3, a particle moving with speed //v// along a a curved path of radius //r// has an acceleration component a = v 2 /r toward the center. As with any acceleration, the net on an object around a curved path is in the direction of the acceleration. In this case, the force is called the centripetal force and it is in the direction of the centripetal acceleration (toward the center). The centripetal force is always perpendicular to the direction of motion. It's usually due to a string, spring, normal force, or friction. It could even be due to a force like gravity (this is why things orbit in space). Examples:



__[]__.



__[]__ In this example, the normal force of the track and the force of gravity is what causes the cart to travel in a circular path. As you can see, at the top of the track, the force of gravity and the normal force both work in the same direction to work as the centripetal force. However, when the cart is at the bottom, the forces are in opposite directions so the normal force has to be much greater to counter the force of gravity and continue the circular motion. This is why you feel "heavier" at the bottom of a loop.   The swings and the Gravitron are both everyday examples of circular motion. In the swings, the tension acts as the centripetal force. In the Gravitron, the normal force acts as the centripetal force.

__**Banked Curves With Friction**__ Motion along curved paths gets a little trickier when you start including banked curves, but when breaking down all the forces and drawing free body diagrams, it becomes quite easy. An example of a banked curve problem is when discussing a car going around a banked turn like the one below:



When talking about a banked turn with friction, we must first decide whether we want to find V max, V min , or V ideal. This choice will determine the way the friction faces or if there is any friction at all. At V max, the car will want to slide out of the bank and friction will point toward the center to keep the car in. At V min, the car will want to slide toward the center but friction will point outward to keep the car on the track. Finally, V ideal means no friction is needed at all. For simplicity sake, we will only find V max right now. The everything else can be found using the same steps but with friction facing different ways accordingly.

**__Drag Forces__** When an object moves through any fluid such as air or water, the fluid exerts a drag force that opposes the motion of the object. Unlike friction, the drag forces increases as the object's speed increases. At low speeds, the drag force is proportional to the velocity, while at high speeds it is proportional to the square of the velocity. Example:

An object is dropped from rest and is falling under the influence of gravity. Thus, the equation we would use to find the acceleration would be

∑F = mg – bv = ma

where b is the drag coefficient. Another important concept to understand is that a drag force puts a threshold on an object's velocity. Once mg = bv, there is no longer any acceleration because the net force becomes zero. Consequently, the object reaches what is called terminal velocity. If you have ever heard the term "aerodynamic", this is the term used to describe an object with a low drag coefficient and high terminal velocity. Racecars and jet planes are just two examples of objects that are designed to be aerodynamic and cut through the wind.



__**DRAG FORCE LAB**__ In this lab I decided to drop coffee filters from approximately 3 meters above the ground. While doing this, I used a motion sensor to calculate the velocity. At some point in the voyage to the ground, the filters reached terminal velocity due to the resistance of air. Once i knew the mass of the filters and the terminal velocity, I could calculate the drag force by setting the weight component equal to bv. I repeated this process increasing the number of filters and thus the weight component each time. (m/s) || calculated drag force || __**Sample calculation**__ __For 1 filter__ .001(9.8) = 1.825b b = .0054 __**Analysis**__ Overall, the results were very good. As the weight component was increased, the terminal velocity increased fairly proportionally to counter this weight increase and keep the drag force somewhat constant. The one outlier was the single filter which gave a drag force of about .001 less than the others. This is most likely due to the fact that the mass was so little causing the filter to flutter and the sensor to read slightly more inaccurately than it did for the other runs. In the end, what this lab proved was that the drag force is constant as long as the object maintains the same shape and is dropped through the same "fluid". It also showed that as the mass increases, the terminal velocity also increases because the drag force has less affect on it. **__Practice Problems__**
 * # of coffee filters || mass || terminal velocity
 * 1 filter || 1 g || 1.825 || .0054 ||
 * 2 filters || 2 g || 3.125 || .0063 ||
 * 3 filters || 3 g || 4.867 || .0061 ||
 * 4 filters || 4 g || 5.902 || .0066 ||
 * 5 filters || 5 g || 7.522 || .0065 ||






 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
 * Content || 4 || 20 ||
 * Organization || 4 || 20 ||
 * Accuracy || 4 || 20 ||
 * Appearance || 4 || 20 ||
 * Participation || 4 || 20 ||
 * TOTAL || 100 ||