Unit 4

Rotational speed is the number of rotations per amount of time. If a lady bug was sitting any where on an LP, it would always have a rotational speed of 33 1/3 rpm.
Tangential speed depends on the radial distance, or the distance from the axis of rotation.
Real life applications of these concepts can be seen in train wheels and gears. Gears work by moving at the same tangential velocity and different rotational velocity. Train wheels self correct by having the same rotational velocity but different tangential velocity.

Rotational inertia is how resistant something is to rotation and it involves the distance from the axis of rotation. If there is a meter stick with two weights taped very near the middle, it would rotate easier because it would have a short distance to the axis of rotation and therefore a lower rotational inertia. If there is a meter stick with two weights taped towards the end, it would be more difficult to rotate because it would have a long distance to the axis of rotation and therefore a higher rotational inertia.

Conservation of angular momentum  means that the torques are equal in a situation such as a figure skater extending their arms and then curling up. When a figure skater is extended, she moves much slower than she does when curled up. Despite this, the angular momentum is the same before and after because when extended, the skater has a small rotational velocity and a large rotational inertia and when curled up, the skater has a large rotational velocity and a small rotational inertia.

Torque causes rotation. Torque = force x lever arm (distance from the axis of rotation). A large force, a large force, or both, create a large torque. For instance, if someone was having a hard time using a wrench, they should get a longer wrench because it would increase the lever arm and therefore create more of a rotation. When something is balanced, its clockwise and counter clockwise torques are equal.

The center of gravity is the average location of mass in an object. The base of support is like the bottom of a box or someone's feet. When the center of gravity is above the base of support, there is no torque/rotation. Lovering the center of gravity makes it more difficult to rotate the center of gravity outside of the base of support. Making the base of support lower creates more stability as well.

Centripetal force is an inward seeking force. For instance the moon stays in orbit around the earth because of the centripetal force pulling it to earth and the fact that it wants to keep going straight.

Mass of a Meter Stick Challenge

Our goal was to find the mass of a meter stick using only a meter stick and a 100g weight.
In order to do this, we placed the weight on the very end of the meter stick and then balanced the meter stick on the edge of a table. When something is balanced, it means that its clockwise and counterclockwise torques are balanced. Because torque = force x lever arm, the force x lever arm of the right side equaled the force x lever arm of the left side.
On the left side we have the 100g weight. The force of this will be the mass of this weight multiplied by the force of gravity, 9.8. The force on the left is 980. The meter stick was balanced at the 30 centimeter point making the lever arm 30 cm.
On the right side, we do not know what the force is. The lever arm is from the point of balancing to the center of gravity of the entire stick which is at 50 cm. The lever arm of the right is 20cm long.
Our equation is 980 x 30 = x x 20. We solved using algebra to get...
1470. the decimal point must be moved over to convert it back into grams.
The mass is 1.47g.

Center of gravity + torque

Although the Leaning Tower of Pisa is leaning it does not fall down. This is because the tower's center of gravity is over, or within the base of support. Center of gravity is when gravity acts on a center of mass. The center of gravity is the average position of mass in an object.
In the picture below, the pink dot is the center of mass, the pink line is the center of gravity and the green is the base of support.  Although the object is leaning it is not falling over because its pink center of gravity is within the green base of support. In the second picture the pink center of gravity is not within the green base of support. Therefore the object will have a blue force resulting in an orange lever arm. A lever arm is the distance from the axis of rotation, or where the object rotates from. When something falls it rotates. In order for something to fall, torque is required because it causes rotation (torque=forcexleverarm). In order to have a large torque, the object must have either a large force or a large lever arm or both. The larger the torque, the larger the rotation. In the situation below, the object falls because its center of gravity (pink) is outside the base of support (green), has a lever arm (orange), and therefore has a torque which causes the object to fall.  

 

Unit 3

Newtons 3rd Law + Action and Reaction Pairs
   Newtons 3rd Law states that every action has an equal and opposite reaction. For instance if a book was sitting on a table, the book would be pushing down on the table and the table would be pushing up on the book. These are called action and reaction pairs. Another example is walking. As a person pushes the ground backwards with their foot, the earth pushes foreword.
Tug of War
   We can use our knowledge of Newtons 3rd Law and Action and Reaction pairs to win a game of tug of war. In tug of war, the amount each team pulls actually doesn't make a difference because of Newtons 3rd Law. What does matter is how hard each team pushes on the earth. If one team pushes back the earth with a force of 5 making the earth push back with an equal force, and the other team pushes the earth with a force of 10, the second team would win.
   We can also use our knowledge of Newtons 3rd Law and Action and Reaction pairs to understand how a horse and buggy move. Similar to the tug of war situation, whatever force the horse pulls foreword, the buggy will pull back in the opposite direction because of Newtons 3rd Law so what makes the buggy move is the horse pushing the ground harder than the buggy pushes the ground.
Forces in Perpendicular Directions
   A person is in a sled on a snowy hill. They slide down the hill. This happens because of forces that occur in perpendicular directions. The weight of the person on the sled is called F gravity. The perpendicular force to this is called F support. If we add up the vectors, we get a force that would bring the sled down the hill. If the F friction is greater than the force that makes the sled move, the sled would stay still. We can use these same ideas in relation to sail boats, currents, and other movements.
   A heavy box is being suspended by a rope which is attached to the ceiling however, one side is longer and has a smaller angle. When drawing in support vectors in either direction we can draw tension vectors. The longer the vector, the more tension there is.  
Momentum and Impulse
   Momentum, or p, is mass, or m, times velocity, or v. The equation for momentum is p=mv. So if a box was being pushed at a velocity of 10 and it had a mass of 5 its momentum would be 50. Impulse, or J, is how much force, or F, occurs over an amount of time, or t. The equation for impulse is J=Ft. We use impulse when talking about something changing momentum like stopping: going from moving to not moving. Continuing off of the box example, if the box stopped, it would go from 50 to 0 no matter how it stopped. This equation is J=change in momentum. If the box was stopped on a concrete wall versus a sponge wall, there would be very different outcomes. To deal with this type of situation we use the J=Ft equation. In this equation, Force is inversely proportional to time meaning if time is small, force is big and vice versa.  If the box stopped on a concrete wall it would be a very abrupt halt = a small amount of time = big force. If the box stopped on a sponge wall it would be a much gentler halt = long time = small force = less damage. This is the reason we have air bags in vehicles.  
Gravity and Tides
   The reason that tides occur is because of the moons pull on the earth. Tides on opposite sides of the earth are always the same. If you picture the globe, the left and right sides would have high tides and the top and bottom would have low tides or vice versa. This happens because of the difference in force felt by each side. For example, one side of the earth, lets call it side A, is adjacent to the moon and the other side of the earth, lets call it side B, is opposite to the first side. Side A would be a shorter distance away from the moon and because of the formula F=m1m2/d^2, it would have a large net force. Therefore, side B would be a larger distance away from the moon and have a smaller net force. The moon would also have a pull on the center of the earth which would be less than the force of side A, but greater than the force of side B. For example, the force of side A is 15, side B is 5 and the middle is 10. As stated earlier, the difference in force felt by each side is what makes opposing sides of the earth have equal tides. To find this difference, we subtract 10, the middle number from 15 and 5, each side. When doing this, we get 5 and -5. This means that we have a force of 5 pulling to the right and to the left. This creates a tidal bulge. Without a difference in force, we would get a net force of 0, creating no tides at all.  High and low tides alternate and occur about every 6 hours with each occurring 2 times a day.  High and low tides occur every 6 hours because of the time it takes the moon to orbit earth. There are also tides called spring tides and neap tides. Spring tides occur when the sun, earth, and moon are lined up either sun, moon, earth or sun, earth, moon. When this happens, there is either a full moon or a new moon and the tides are unusually high and unusually low. Neap tides occur when the sun, moon, and earth do not line up either sun, earth, and moon above or below the globe. When this happens there is a half moon and the difference between the tides are unusually low.
Conservation of Momentum
   Because of what we know from Newtons 3rd Law, momentum is always conserved. When playing pool, one ball hits the other resulting in the ball that was moving to stop and the ball that was still to move with the same speed as the first one originally moved. A ball could also crash into another cart resulting in them both moving together. 

Tides

This video is a time lapse of the Bay of Fundy's tides. Although this is an extreme case of high and low tides, it clearly shows the difference between high and low tides.
The reason that tides occur is because of the moons pull on the earth. Tides on opposite sides of the earth are always the same. If you picture the globe, the left and right sides would have high tides and the top and bottom would have low tides or vice versa. This happens because of the difference in force felt by each side. For example, one side of the earth, lets call it side A, is adjacent to the moon and the other side of the earth, lets call it side B, is opposite to the first side. Side A would be a shorter distance away from the moon and because of the formula F=m1m2/d^2, it would have a large net force. Therefore, side B would be a larger distance away from the moon and have a smaller net force. The moon would also have a pull on the center of the earth which would be less than the force of side A, but greater than the force of side B. For example, the force of side A is 15, side B is 5 and the middle is 10. As stated earlier, the difference in force felt by each side is what makes opposing sides of the earth have equal tides. To find this difference, we subtract 10, the middle number from 15 and 5, each side. When doing this, we get 5 and -5. This means that we have a force of 5 pulling to the right and to the left. This creates a tidal bulge. Without a difference in force, we would get a net force of 0, creating no tides at all.  High and low tides alternate and occur about every 6 hours with each occurring 2 times a day.  High and low tides occur every 6 hours because of the time it takes the moon to orbit earth. There are also tides called spring tides and neap tides. Spring tides occur when the sun, earth, and moon are lined up either sun, moon, earth or sun, earth, moon. When this happens, there is either a full moon or a new moon and the tides are unusually high and unusually low. Neap tides occur when the sun, moon, and earth do not line up either sun, earth, and moon above or below the globe. When this happens there is a half moon and the difference between the tides are unusually low.
In the summer, my family likes to visit our friends in Rhode Island. They own a house on the Sakonnet beach. Here is a link to a Sakonnet tide chart: http://ri.usharbors.com/monthly-tides/Rhode%20Island/Sakonnet . Right now, as I am writing this post at 8:50 pm, the beach is in-between high and low tides and turning into a high tide and experiencing neap tides. 

Newtons 3rd Law Resource

Despite the minimal animation, I think this video does a good job simply and clearly explaining Newtons 3rd Law. It helps to explain action and reaction pairs and clearly states the law. I found the bicycle example most helpful because it showed a real life application of the law.

Newtons Second Law Unit Summary

    In unit 2...

 We learned about three different types of free fall: free fall straight down, free fall thrown upward, free fall with projectile motion, and free fall straight down with air resistance.

     1) The simplest form of free fall would be, for example, a ball was dropped off of a building. In this situation, air resistance is negligible and the object is falling from rest. Free fall is when objects fall due to the acceleration of gravity only.
     Say you were on a hike. You come across a cliff. You want to know how far down the drop is. In order to do this, you could drop a rock off of the cliff and time how long it took to make the fall. In order to calculate the hight of this cliff, you need to know a formula: d=1/2gt^2. In this formula, g or the force of gravity, will theoretically always be 10 (but in the real world it is 9.8). Say it took the rock 8 seconds to fall. You would say d=1/2x10x8^2. If you solve this equation, you would find out that the distance, or hight of the cliff is 320m.
     If you want to know how fast the rock, or whatever object you drop from whatever hight is moving at a certain time, you need to know the equation: v=gt and remember that because the force of gravity is 10, our velocity will increase by 10m/s every second. So say you want to know how fast the object was moving after 5 seconds of falling. You would simply multiply 10 (gravity) by 5 (time) and you would get the velocity at that time. In this scenario, the velocity would be 50 m/s.

     2) You are watching a soccer game. The goalie catches the ball and decides to punt it down the field. She tosses the ball up, waits for it to come back down, and then kicks it. You can figure out how high the ball goes, how long it is in the air, and how fast it is moving at any time. As you can see in the picture, this speed will decrease by 10m/s every second so by drawing out what we know, we can figure out the aforementioned three things just by knowing the starting velocity or the time in the air.
     If we want to know high the ball was at the top of its path before it started to fall, we use our trusty d=1/2gt^2 formula. So in this case, we have figured out that the ball was in the air for 4 seconds before reaching the top of its path. If we plug that number in for t, we find that it reached a hight of 80m. Lets say we want to know the balls hight at 2 seconds. In order to find this, we need to find the total hight as we did before (pictured in green) and subtract this from the distance pictured in blue. This equals the orange hight, or the hight at 2 seconds. 

     3) Once the goalie kicks the ball, it is propelled foreword and shoots down the field. This is an example of projectile motion. The goalie kicks the ball 50m down the field at a 45 degree angle and the ball stays in the air for 4 seconds. You can figure out how hard she kicked the ball. As you can see in the picture, you can predict where the ball would be every second and how fast it would be going at each second. If we look at our picture, we can see that the ball was kicked at a vertical speed of 20m/s at 0seconds. Remember that its horizontal velocity is constant. In order to find it, we just plug our numbers into the formula: v=d/t. In order to figure out how fast the ball was actually moving at any given time, we draw out a picture like this:   The actual speed will be the hypotenuse. In order to solve for this, we just use the pythagorean theorem: a^2+b^2=c^2, or one of our special triangles: 3, 4, 5, or x, x, xsqrt2. Remember that the square root of 2 is 1.41. So we can find out that in order for the soccer ball to land 50m away, the goalie must kick the ball at about 22 m/s. We can figure out how long the ball will be in the air, how fast the ball will be at the top of its path, and how far away it will land. Remember that the vertical velocity is the main component. In summary, the big formulas to remember are: vertical: d=1/2gt^2, v=gt horizontal: d=vt, v=d/t.

     So lets say that for some strange reason, the goalie who is holding a ball sprouts wings and begins to fly foreword at a speed of 90m/s 125 meters above the ground. She wants to make a goal by dropping the ball into it. The ball has a constant vertical acceleration and a constant horizontal velocity. First lets find out how long the ball will be in the air by using d=1/2gt^2. We should get 5s. Remember we have to drop the ball very early because the ball is moving foreword at 90m/s and will continue to because of inertia. In order to find how far away to drop the ball in order to make a goal, we use the formula v=d/t and find out that horizontal distance is 450m. In order to find where the ball would be in the vertical direction each second, we use d=1/2gt^2. In order to see the actual estimated path, just find where the horizontal and vertical would meet up.

      4)In this unit we also learned about Newtons Second Law. The law states that acceleration is directly proportional to force and inversely proportional to mass. This statement can also be written as a formula: a=F/m or a=Fx1/m. This means that if acceleration increases, force would increase or if acceleration decreased, force would decrease also. This also means that if acceleration increases, mass decreases or if acceleration decreased, mass would increase. In the real world, Newtons Second Law can be seen in a person pushing a box. If the box was light (or had a small mass) it would be easier to move, or accelerate and if the box was heavy (or had a large mass) it would be more difficult to move or accelerate. If you had a box and pushed it just a little bit (with a small amount of force) it would not accelerate quickly. If you pushed the same box with a lot of force it would accelerate quickly.

     5) We did a lab to demonstrate these concepts and illustrate how acceleration depends on force and mass. In this lab, we had a cart on a track with a string attached. The string ran over a pulley to a hanging weight below. In this example, the hanging weight applied the force that caused the acceleration. We needed to find all of the components in our formula: a=F/m. In this case, the acceleration is basically how fast the cart goes. To find the mass, we added up the masses of the cart and hanger to find the total mass of the system. To find the force, we found the weight of the hanger by using the formula w=mg (weight equals mass times force of gravity) and kept this constant throughout the experiment. In .10 kg increments we added more masses onto the cart therefore changing the the total mass of the system. We found that as the mass of the cart increased, the acceleration decreased. In our next part of the experiment, we kept the total mass constant, but moved around the weights from the cart to the hanger one at a time. We found that as the force of the cart increased the acceleration increased.

     6) In this unit we also learned about skydiving. Skydiving is like free fall, but weight matters, and air resistance is a large factor. As a person jumps out of a plane, they are pulled down by the F-weight (which is found using the formula w=mg) and they accelerate towards earth. F-air (the force of air resistance) acts in the opposite direction and increases as the person gains speed. This is because acceleration is directly proportional to force. In this situation the force is air resistance and acceleration is how fast their velocity is increasing. F-air increases until it becomes equal to the constant F-weight. This is called terminal velocity. In terminal velocity, the net force is 0 which means the person is no longer accelerating (although they continue to move downward) and is in equilibrium. Once the person opens their parachute, the F-air becomes much greater than before, the person is no longer in equilibrium, and the speed of the person slows down. Because the speed is decreasing, air resistance decreases also because they are directly proportional. These two factors continue to decrease until F-air is equal to F-weight again. This is the second terminal velocity. In the second terminal velocity, the speed is slower than in the first and the air resistance is larger than in the first. The person continues to fall toward the ground at this much slower speed and can safely land on the ground. VIDEO HERE!!
     
 F-air and F-weight are important parts of falling things besides sky diving.
     If you dropped a crumpled piece of paper and a flat piece of paper, although they are the same weight, the crumpled paper would land first. This is because of surface area. The crumpled piece of paper has a small surface area, so it would have to accelerate longer in order to reach terminal velocity and get its F-air to equal its F-weight. The flat piece of paper has a large surface area, so it would not have to accelerate long to reach terminal velocity and have an equal F-air and F-weight.

     If you dropped a lead ball and a ping pong ball, although they have the same surface area, they would land at different times. This is because they have different weights. The lead ball has to accelerate for much longer in order to reach terminal velocity and get equal F-weight and F-air and the ping pong ball doesn't have to accelerate long to reach terminal velocity with equal F-weight and F-air.

     It was interesting to learn how physics plays a role in a wide range of things as simple as tossing a ball into the air, as common as playing sports, and as exiting as skydiving.