Monday, December 9, 2019

"If you can dodge a wrench you can dodge a ball"

Dodgeball is an interesting game which has a lot to do with physics. The force the person uses to throw the ball and the actual trajectory of the ball itself are all regulated by the physics of the world. When you play dodgeball, you want to try and avoid 'lobbing' the ball. A faster trajectory is preferred so as the person you're aiming for does not catch the ball - otherwise you're out. Therefore, when considering how to throw the ball there are certain forces that need to be accounted for.

First, the throw of the ball involves a lot of physics. The wind up to the throw involves the use of many muscles to not only move the arm but to gain momentum to throw the ball with a force. Since muscles are very complicated and the wind up for a throw is not a simple motion, I will present a simpler version of a throw to explain some of the physics behind a throw.

This image shows White Goodman gearing up to throw the dodgeball. For this example, we will say he is only throwing using his forearm (keeping his elbow at 90 degrees). In this example it is a torque problem. There is the force that the arm exerts on the ball to hold onto it (centripetal force) and the torque of the ball. When Goodman lets go of the ball, the total force that he exerts on the ball will be equal to the instantaneous torque of the ball at the moment of release.

When the ball is in the air gravity and air resistance are the two forces acting on the ball.
The free body diagram for the dodgeball shows the forces acting on the ball. As the ball travels through the air, it will slow down due to the air resistance acting against the trajectory as well as the force of gravity will bring the ball down to the ground.

Lastly, assuming the person doesn't catch the ball, the hit of the ball against the target represents a perfectly elastic collision. In this case both momentum and kinetic energy are conserved, and the force the ball exerts on the person can be calculated.

Overall, the physics of dodgeball involve many different concepts including torque, kinematics, and collisions. Lastly the quote from the movie Dodgeball states "if you can dodge a wrench, you can dodge a ball." It is important to note that each of these objects have different masses and rotational patterns, so it might not be the best idea to practice or play dodgeball with wrenches. Actually, just don't throw wrenches.

Structural Integrity of a Gingerbread House

As the holiday season approaches, gingerbread houses are again becoming a popular activity with my friends. After fall break, we made one that from Trader Joe's that was Halloween-themed and in the shape of a pentagonal prism (see below; very spooky).  It was really fun to put together, but when we were walking it to another friend's apartment, it sadly fell to pieces. This past weekend, we made another gingerbread house that was triangular-prism-shaped and holiday-themed. When making it, we accidentally watered down the icing sugar too much, but others kept going with the construction, and surprisingly, even without the icing, the house held together when we tried to move it, arguably better than the Halloween-themed one. In the past, I have only ever seen gingerbread houses in the traditional house shape with a rectangular base and a triangular roof, so I had no idea that they even made triangular gingerbread houses! We always say that for gingerbread houses, structural integrity is key, and the components of a structurally-sound gingerbread house can be explained using physics.

I began by drawing the free body diagrams for the two houses. I have drawn the forces of gravity (blue arrows) acting at the centers of mass (green dots) to illustrate the point, though they act along the whole length of the gingerbread slabs in reality. In addition to the gravitational forces, there is also friction (red arrows) holding the pieces together, preventing collapse. Though I have not shown them, there is also friction between the roof pieces and the front and back panels in both cases.

On the left is the Halloween-themed gingerbread house, with 4 major parts comprising its exterior. On the right is the holiday-themed gingerbread house with only two slabs making up its outer sides. By comparing these drawings and the general structures in the pictures above, it becomes apparent that there are significantly more forces acting in the Halloween-themed house than for the holiday one. Also, the forces in the Halloween-themed house are distributed vertically, making for greater gravitational potential energy being stored in the roof pieces of the house than in the holiday-themed one. Similarly, the whimsical tapered shape of the house with a smaller base also makes the house itself more unsteady, since the center of mass of the whole structure would be elevated in comparison to the holiday-themed one. As a result of these, even very basic consideration of physics could have predicted that the Halloween-themed house would be structurally unstable (but still very cute). In this way, we can learn from the lessons of this gingerbread calamity for future designs.

When imagining the most structurally stable gingerbread house, the isosceles triangular prism is the ideal shape, because it can distribute its gravitational forces along the gingerbread evenly, while keeping the its center of mass at its base, and minimizing the height of the house (reducing gravitational potential energy). Also, in terms of construction, it would be better to use thicker gingerbread compared to the thin and cracker-like gingerbread found in some kits. This change would provide better contact between the pieces to create a larger surface area for friction to act, thus, holding the house upright. (One could even go so far as to file down the ends of some pieces to create flat edges, which would also help increase surface area at these vertices.) Finally, to help with the friction and contact, a more viscous, somewhat hard-setting icing would be ideal since it would be able to fill in any cracks or gaps between the gingerbread pieces to increase the surface area experiencing friction with the gingerbread, and increase the coefficient of friction for these surfaces. With that, I hope this analysis can help you with any future gingerbread endeavors, and happy holidays!

The Physics of Shopping

With one of the year’s most popular shopping days behind us, I decided to reflect on the physics of shopping to see exactly what physical toll those Black Friday deals take on our bodies.  

To clearly paint this picture, let me narrate an example story. To prepare for the winter snowstorm making its way towards Colgate, you decide that you want to buy a new sweater, pants, winter coat, hat, pair of gloves, scarf, and high boots to help you bundle and keep warm. 

You enter the store walking at an average human walking speed of 3 mph (1.34 m/s) when you see THE perfect winter sweater hanging on the rack and increase your speed to an average running speed of 2.62 m/s in just 3 seconds. This means that your acceleration is 0.43 m/s^2, requiring a change in kinetic energy equal to 195 J. 

3 mi / 1 hr * 1609 m / 1 mi * 1 hr / 3600 s = 1.34 m/s
1 mi / 10.21 min * 1609 m / 1 mi * 1 min / 60 s = 2.62 m/s

a = (vf - vi) / t = (2.62 m/s - 1.34 m/s) / 3 seconds = 0.43 m/s^2

KEi = 0.5mv^2 = 0.5 * 77 kg * ((1.34 m/s)^2) = 69.1 J
KEf = 0.5mv^2 = 0.5 * 77 kg * ((2.62 m/s)^2) = 264.3 J
Change in KE = KEf - KEi = 264.3 J - 69.1 J = 195 J

Therefore, being an average American female weighing 77 kg and standing 1.65 m tall, you need to exert a force of 33.1 N in the x-direction to get that sweater before someone else does. 

F = ma
F = 77 kg (0.43 m/s^2)
F = 33.1 N

The average masses for these items you purchased are as follows: 0.5 kg for a sweater, 0.4 kg for pants, 3 kg for a coat, 0.2 kg for a hat, 0.15 for gloves, 0.3 kg for a scarf, and 1.8 kg for boots. Although you are environmentally conscious, you forgot to bring your reusable shopping bag and had to collect your new items in a plastic shopping bag that has a mass of 0.0055 kg.  

Based on these measurements, your new purchases achieved a total mass of 6.36 kg. Therefore, if you are holding this bag straight at the side of your thigh (ignore slight angle from how the bags project), it exerts a force of about 62.3 N downward. 

m = 0.5 kg + 0.4 kg, 3 kg, 0.2 kg, 0.15 kg, 0.3 kg, 1.8 kg = 6.36 kg

F = ma = mg
F = 6.36 kg (9.8 m/s^2)
F = 62.3 N 

Therefore, according to Newton’s First and Third Law of motion, your arm muscles need to exert 62.3 N of force in the upward direction to keep the bag at rest at your side. 

Maybe Cyber Monday online shopping seems like a much better idea for your arm (not so much your bank account though)!


Saturday, December 7, 2019

Physics of Running

Physics of my Knee’s While Running

Throughout my life, I have been an athlete. Whether it’s team sports like soccer and basketball or individual sports like tennis and gymnastics, I have been active my entire life. These various activities have caused a variety of sports-related injuries in my life. Some of my injuries were so extreme that they caused my muscles and joints to compensate for pain by moving in new ways. An example of this is the way my hip and knee joints work while I run long distances. Instead of the 2D motion that most legs make while running (Figure 1), my knees act more like a ball in socket joint that can rotate in 3D such that my steps are in front of one another, but my tib-fib makes a whipping motion to the side between strides–similarly to Figure 2. As the knee is a hinge joint, this motion is very slight and therefore most people do not notice exactly what is off when they see me run from the side; they just can tell that my stride looks different. This altered motion is a result of my hips–an actual ball in socket joint–that rotate my femurs medially. This rotation causes my lower legs to get in the way of each other unless this whipping motion occurs. Overall this change in joint function causes my running speed to be decreased as the energy I put into my stride partially goes into the rotational motion of my knee joint.  

Figure 1- 2D running motion

Figure 2- Ball in Socket Joint Rotation

The Physics of Baseball Bats

Aspiring baseball players across the nation are preparing for the spring season to begin and many of them are freshmen about to try out for their high school team. This time is a stressful one where dreams are achieved or broken and many players are shopping around for new gear to aid their performance. In my opinion, the most important choice one can make about their baseball gear is the bat they decide to purchase; while the majority of bats generally serve the same purpose, the construction of the bat can either enhance or hinder the ability of the player. Under the national BBCOR league, bats that are created by companies must perform about equivalently to the wooden bats seen in the MLB; however, clever engineers have discerned a way to tailor a bat to the specific needs of the player. Many baseball sites will display a graph similar to the one below:

This chart is an incredibly simplified explanation about the moment of inertia (MOI) of the bats a site has for sale. A baseball player transitioning from another league can use this chart to decide which baseball bat would be best for them. The numbers in the chart inform the player how "heavy" the bat will feel to swing; heavy is in quotes because, as a BBCOR standard, all bats will be -3. This means that the bat will weight (in ounces) three less than the length of the bat (in inches). Therefore, every 33" bat will weigh 30 oz. If all the bats weigh the same, how do the numbers in the chart come into play? The answer lies in physics.

The distribution of mass throughout the length of the bat is the distinctive feature demonstrated in the chart. The Mass Index shown above gives the player an indication of the MOI of the bats. For example, a lower number (650-700) means that more of the mass of the bat is positioned closer to the player's hands. This diminishes the distance the mass is from the axis of rotation, making the bat feel lighter and making it easier to swing. A higher number (950-1100) means that more of the mass is positioned near the endcap of the bat (the furthest position on the bat from the player). This makes the bat feel heavier and making the bat more difficult to swing. Younger and smaller players that are not strong enough to effectively use an "end-loaded" bat would benefit from the bats with a lower Mass Index (MOI) to create adequate bat speed through the zone to make consistent contact with the pitch. Conversely, a stronger, more developed player that can swing a bat quickly through the zone can use a bat with a higher Mass Index to increase the angular momentum behind their swing, giving them more power and launching the ball further.

The Physics of Planes

Over the past two weeks, I have been on six flights and with two more coming up this week. After spending so much time in the air, I began to think about the logistics of a plane, or rather, the physics behind flying.

First, let's talk about propellers. A propeller basically lifts an airplane forward. For ease of understanding, a propeller is similar to a wing, but it spins. Similar to a wing of a plane it produces lift in a forward direction. This force is called thrust. The propeller works with the help of the engine.

Thurst = F = A x (delta p)

The rotational motion of the propeller while through the air creates a difference in air pressure between the front and back of the blades of the propeller. This increases the air pressure on one surface and decreases the air pressure on the other. It is also important to know that the pressure over the top of the wing as the plane is lifting itself into the air is greater than the pressure below the wing.

Through Newton's Third Law of motion, for every action, there is an equal and opposite reaction, We can see how the propeller pushes or pulls you forwards by pushing a mass of air behind you. The combination of the propeller, engine, calculated momentum, thrust, velocity, and pressure, pilots can keep huge planes in the air with 100s of people and cargo in them. The next time you are on a plane, see if you can see the propeller rotating as you fly as well as the flaps on the wings opening and closing to account for air pressure change. I definitely will.

The Physics Behind Pro Wrestling

Pro wrestling, perhaps the greatest entertainment event ever created by man. We see moves that should be impossible pulled off on the big screen, but what's the physics behind them? Let's explore a prominent pro wrestling move to analyze the physics of it, and whether it'd be effective or not.

To start with a classic: The German suplex. A wrestler grips another wrestler from behind and lifts  them behind their head, slamming them into the ground. In this case the wrestler is acting as a lever arm and producing the torque  required to move the opponent up and  over their  head to the  ground. The tricky part of a German suplex is  the balance between power applied and power delivered. Depending on the grip, if the opponent is farther from the torso more torque is required to move the opponent up and  over your  head, however, more force is applied when the opponent strikes the ground. If the opponent is closer to  your torso less torque is required to move the opponent up  and over your  head, but less force is delivered on the downward strike. Therefore, a "sweet spot" exists where the opponent is the perfect distance from your torso where you possess necessary strength to heave them over your head and deliver a  strike with maximum force.