Unit 4 Summary

This Unit we learned about rotation.

Rotational and Tangential Velocity:

When speaking about rotation, there are two different kinds of velocity. There is rotational(or angular) velocity, and tangential(or linear) velocity.

Rotational velocity: measures how many times something rotates in a given unit of time. 

Tangential velocity: measures distance over in a given unit of time. This is the kind of velocity we were already familiar with. 

There are many situations where these two seem similar but are different. For instance when two children are riding on a merry go round, one on the outermost horse, and one on the inner most horse, their rotational speed is the same. If you can imagine, they rotate the same number of times per minute. However, the outermost child must cover more distance and therefore has a higher tangential velocity. 

Rotational Inertia:

Regular Inertia represents something's resistance to change.

Rotational Inertia: Rotational Inertia is specifically an objects resistance to change rotation. 

Inertia in regular circumstances is usually represented as mass. With rotational inertia it refers specifically to the distribution of the mass. When the mass is distributed further from the axis of symmetry, the rotational inertia increases. Be careful to not mistake this for increased rotational velocity. Inertia is resistance, so when the inertia increases, the rotational velocity decreases. The converse is true for when the inertia decreases. 

An Example of this is when an ice skater spins. When the ice skate pulls their arms(mass) in closer to their axis of symmetry, their rotational velocity increases. And when they extend their arms, they slow down. 

Conservation of Angular Momentum: 

We know very well that momentum is conserved. That is total momentum before = total momentum after.
The switch to angular momentum is ridiculously simple: angular momentum before = angular momentum after. 

And Rotational Momentum is also similar to normal momentum: 

Angular Momentum = (Rotational velocity x (Rotational Inertia)

This principle of conservation simply states that, for example the ice skater, would have the same angular momentum when she pulled her arms in as when she extended them out. This is because of the way inertia and velocity act opposite to each other. When inertia increases, velocity decreases and vise versa. They are always balancing each other out. 

Torque:

Torque is the thing that causes the rotation.

Torque is made up of two components: Force and Lever arm. 
Torque = Force x Lever Arm 

For instance in this picture, the side of the see saw with the girl on it has a greater Torque because she has a greater force, so it rotates with a counterclockwise torque.

Each rotation has a clockwise torque and a counterclockwise torque. 

When something is balanced, the two torques are equal. That is, 

Clockwise Torque = Counter Clockwise Torque

Here is an example of a meter stick that has been balanced on the edge of a table with a 100 gram weight on one side and the calculations to prove it is balanced:

*Remember that the Force is the Center of Gravity. The Lever Arm is the distance from the Center of Gravity to the axis of rotation. 

Center of Mass/Gravity:

To keep balanced you must keep your Center of Gravity over your base of support.

For instance, The leaning tower of Pisa is tilted, however, it's Center of Gravity is still over it's base of support.

This is why football players keep their feet further apart, because widening your base of support makes it easier to keep your Center of Gravity over it. And they are less likely to fall over. 

When wrestlers bend their knees, they do so to lower their center of gravity closer so that is, again, harder to get it out of the base of support, and thus harder to knock them over. 

Centripetal Force: 

Centripetal Force is a force that goes inwards from an object to the center as it rotates. 

This is why it is also called a center seeking force. 

It is a combination of an object that is already moving forward and the force of gravity downwards. 

Here is a picture of centripetal force:

(ps. sorry this is the lamest diagram ever...but it was either super simple or super complicated. If you need visual help check out the podcasts.)

Remember that centrifugal force is not real!

And that was Unit 4! 

Mass of the Meter Stick Lab

In this Lab, we were tasked in finding the mass of a given meter stick by using our knowledge of balance and torques. We were told to use a meter stick and a lead weight that weighs 100g. Once we had done our calculations, we would check the mass we got by actually weighing the meter stick and comparing the numbers. So we began....

Step 1:
First we did some demonstrations in order to better our understanding of torques, and to bestow on us the tools we need to find this information(the mass of the meter stick).

Here are the demos we did:

These exercises really helped us figure out the information we needed to gather and compile. We began to learn how to use the information we had to find more information.

Step 2: 

We began to plan after doing these demos. The demos were written in to solidify our ideas about balancing, but the planning was all vocal. We started to set it up and talk through way we might be able to calculate the mass.

Step 3:

Steps 2 and 3 bled together because trying out ways to find the mass were simultaneously our plans for doing them. The planning and the doing were not separate but we did them at the same time.

What we tried was a lot of writing numbers down and figuring out how they fit together. This is what finally worked:

We started by finding the Center of Gravity for the stick without the weight on it by balancing it on the edge of the table. It didn't end up being 50cm, it ended up being 50.3 cm. Then we added the weight to the end and tried to find the Center of Gravity again. That number we found was 28.7 cm.

Then we had to figure out what to do with those numbers. We knew that the meter stick was balancing, and when things are balanced, it is because their clockwise torque and counterclockwise torques are equal. We set up the equation that we set up in the demos to use the lever arms and force on one side to find the force on the other side.

clockwise torque=counterclockwise torque
Force * Lever Arm = Force * Lever Arm

Then we plugged in using .98 N instead of 100g for the weight because it needs to be in Newtons to be a force. We knew that 50.3 cm was still the center of gravity, so we subtracted 28.7 from 50.3 to find the lever arm of the side of the stick resting on the table. We got 21.6 cm, and used it as the lever arm for that side of the equation.

(.98) * (28.7) = (F) * (21.6) 

We found that the force on the right side(aka the entire meter stick, because we used that Center of Gravity) was 1.3 Newtons.
We converted that into mass using the w=mg equation because what we found with the torque equation was the weight of the meter stick. We use that equation to convert it to mass, which is what we are looking for and what we will measure. We found that the mass was 130g.

When we weighed it it was actually 120g. So we were off by ten grams which is kind of a bummer and void of satisfaction. Oh well, we learned about Physics anyway.

Here is a picture of our field notes:

Here is a Drawing of what is going on:

Resource Blog

Torque resource:

This resource is a Khan Academy tutorial/introduction to Torque. I found this video surprisingly informative because I had only seen a couple of science Kahn Academy videos but did not find them incredibly helpful. The ability to draw and animate movement by use of the medium he uses helps the effectiveness of teaching. It does teach the information a little bit differently from the video Ms. Lawrence made which is similar in style which may be confusing, however, it may also be very helpful. I found it helpful.

Link:

Introduction to torque: An introduction to torque


Center of Mass/Gravity resource: 

This page is a blog post on a blog I hadn't heard of but is apparently made by or funded by PBS. This source integrates torque in as well which helps to relate them. The flow of the post is intuitive and east to follow, and it begins with an example to strengthen understanding through real-life. Over all I found this source very helpful.

Link:

http://www.pbs.org/opb/circus/classroom/circus-physics/center-mass/

Unit 3 Summary

1.) Newton's Third Law and Action Reaction Pairs:

 Newton's Third Law: Every action has an equal and opposite reaction.

For instance, a pair of tug of war teams are always pulling on each other with equal and opposite forces. A car and a truck crashing exert equal and opposite forces on each other. You state this pair like this:

Car pushes truck forward
Turck pushes car backward
(be sure to include vectors)

Or a skateboard:

- The effect that makes a difference in the pairs, especially one like the tug of war example, is friction. The greater force of friction allows one team or the other to win. Two things that affect friction are weight and the nature of the surface.

- Many times there are multiple action reaction pairs for every situation. For example, a grapefruit sitting on a table is pairing with both the earth and the table. These pairs are not equal and opposite necessarily, only the individual pairs are equal and opposite.
- Make sure to remember that accelerations in the opposite direction are negative.

This page contains great practice questions for understanding Newton's Third Law.

2.) Vectors 

Vectors are a device that show both direction and magnitude.  
More simply, vectors are arrows. We show magnitude with the length (and possibly opaqueness if you want) of the arrow, and the direction is the direction the arrow is pointing.

Adding vectors is helpful when determining the actual direction something will be traveling and the speed at which it will be traveling, by considering the forces acting upon the object.


A good example is a canoe on a river. If there is a current running south and you are paddling east, using vectors, you can determine where the canoe's path will take.

To add these vectors, draw lines parallel to each of the existing vectors, and then the line of actual direction is from the center to the intersection of the parallels.
Use this method also to help determine tension.

When trying to determine a lesser or greater tension, inverse the parallels so that they intersect the opposite line. The tension is represented by the distance between the center and the points of intersection. Be sure to put direction on the friction lines- I have made this mistake before. 

3.) Tides/Gravitational Force  
The formula for Gravitational Force is F=G m1m2/d^2

This formula explains the gravitational force between masses like planets. 
This equation and the information we can extract from it help explain a lot of things including tides.

Tides are caused by the gravitational force between the moon and the earth. Even though the sun has much (much much much) more mass, the closeness of the moon ultimately makes the force between the moon and earth greater and have a greater effect. 

The equation dictates that distance between the objects and the force are inversely proportionate. Same with mass, so the close distance between the moon and Earth made up for it's small mass.


-The Earth has a tidal bulge around it that is created by the moon's gravitational force and it creates high tides on each side of the moon and low tides on the adjacent sides.  The tides are caused by a difference in force from the moon because the distance is different at the two points relative to the center of the moon. There are six hours between each low and high tide and 12 between high tides and 12 between low tides. So, there are two low tides and two high tides each day. 

- When the moon and sun and earth are in line, the tides are called spring tides which are higher than the normal high, and lower than the normal low.When the sun is not in line it is rather during a quarter phase of orbit, it is called a neap tide. These tides are more lukewarm than the spring tides, in fact they are lower than the normal time tides. 

                                                           

This video is really helpful in understanding tides:

3.) Momentum and the Conservation of it

The momentum formula is p=mv
(the units for p is therefore kgm/s)

p total before= p total after (because of Newton's Third Law)



An important detail about momentum is that the momentum os a system is conserved. That is, 

When we are assessing two objects, like carts, collide, we can calculate the speed after by taking this assumption. Use this equation:

MaVa+MbVb= (Vab)(Ma+b)    - plug in the masses and velocities to find the velocity after the carts collide.

 
3b.) Impulse
Impulse (or J) is the change in momentum.

To change momentum, you need a change in force.
When thinking about changing the Force, two main things matter: 

1. The amount of Force applied  2. Time F is applied

For these things, we explain the equation for Impulse: J=F(change in t)

This equation shows that distance and force react in a certain way to each other. When time increases force decreases and vise versa. 

An Example of this would be landing on a hardwood floor rather than a memory foam mattress. The mattress increases the time of force applied (do not forget that the change in momentum/impulse is the same), which decreases the force. That is why you wouldn't get as hurt on the mattress. 
When answering these problems be sure to mention:

-That the object will go from moving to not moving no matter what
- The relationship between change in momentum and impulse
- The relationship between time and force when talking about impulse

This page helps to connect Newton's Second Law, Newton's Third Law and Change in Momentum in contribution to Impulse. The building blocks are really good to understand so check it out.

Tides

This Video explains tides well: It especially is helpful with the parts about forces on the earth with vectors and explanation. It also has a really cool demonstration of the tides in the program he has. The forces are the most confusing in terms of where the force is in relation to the core and the water on the surface and this video helps with that.



Because the moon is closer to the sun it is the force causing the tides not the sun even thought the sun has more mass and is pulling harder on the earth. The Earth has a tidal bulge around it that is created by the moon and it creates high tides on each side of the moon and low tides on the adjacent sides.  The tides are caused by a difference in force from the moon because the distance is different at the two points relative to the center of the moon. There are six hours between each low and high tide ad 12 between high tides and 12 between low tides.So, there are two low tides and two high tides each day.
There are different kinds of tides according to where the sun is relative to the moon and earth. If the sun is in line with the moon and earth it is called spring tides and it makes the high tides higher and the low tides lower. If the sun is not in line it israther during a quarter phase of orbit, it is called a neap tide.
http://www.tides4fishing.com/us/north-carolina/nags-head

http://www.tides4fishing.com/us/north-carolina/nags-head

Right now the tides are low, but they will be high very soon. 
It is in between but soon it will be neap tides.

Newton's Third Law Resource

The Physics Classroom link on Newton's Third Law:

This source was really helpful in strengthening my understanding of Newton's Third Law. It provides a very conclusive summary of the Law and it's meaning. It would be especially helpful in making the general idea and the abstract of the concept more concrete.

The site also includes examples and sample problems that you can try out and check your answer. I find these really helpful because often I read the text and think I retain the info, but the questions could reveal I don't.

This page on action and reaction pairs is helpful in understanding examples of the law and what they mean for actual objects. It also includes practice questions.

Unit 2 Summary

Unit 2 Summary:

The second unit was about Newton's Second Law:

Newton's Second Law is:
 acceleration is directly proportional to force and inversely proportional to mass.
or:
a=F(1/m)

In the Newton's Second Law lab, we set out to prove this. 
We did so by changing mass and Force to see how it would affect acceleration.

In the first experiment:
-we added mass and found that when we added mass and measured the acceleration, the acceleration decreased. 
-Then we graphed the data and compared the equation of the line to Newton's Second Law. 

y=mx+b
a=F(1/m)

The slope of the line is always the factor we kept constant during the experiment. 
in this case, it was force(we kept force constant by keeping the weight of the hanger constant).

When we compared the slope(theoretical force), to the actual force we measured(the wight of the hanger), we found that they were often within 10 percent of each other, depending on which group you asked. If it was within 10 percent, we can say that it suggests Newton's Second Law is true.

In the second expierement, we repeated the same procedure, but kept mass constant and increased force. We did so by transferring the mass from the cart to the hanger. We found that when we increaced the force, the acceleration also increased.

Then we learned about Free Fall, that is moving without air resistance. 

In free fall, the only force acting on the object is gravity. The force of gravity in free fall is 10m/s^2. 
-When a ball falls, the speed increases by 10 m/s every second. 

-When a ball is thrown up, it decreases every second by 10m/s. 
-at the top of it's path, the acceleration is 0 and it is at equilibrium. 
-When it falls back down, It increases every second again. 
*when calculation for a ball that has been thrown up and has come back down, you can only use the formulas if you measure the path it took starting at 0m/s.

For instance, to find the total time in the air, you must use d=1/2at^2 only for the path from equilibrium(at the top) to the ground. Then you can double the time to get the answer. 

Throwing things up at an angle:

When you throw things up at an angle, you must use the vector as the hypotenuse of a triangle with the other sides being vertical velocity and horizontal velocity. You treat the vertical velocity the same as you would throwing straight up as it increases by 10m/s when falling and decreases by 10m/s when rising each second. The hypotenuse of the triangle is used to determine the actual velocity at each second. 
*the angle may help you determine of the triangle is a special triangle

Projectile Motion:

When measuring projectile motion, you must take into account the vertical and horizontal properties. You must use different equations for each one: 

vertical:
d=1/2at^2
v=at

horizontal:
v=d/t
d=vt

Use each of these to find whatever part of the problem you are asked for, but make sure you are using the right equation. Hight determines time so the time will be the same for both of them.
To find the actual velocity at any given second, you create a triangle with the speed on each side, both horizontal and vertical, and the hypotenuse represents the actual velocity.

Falling Through the Air(Falling with Air resistance):
When falling through the air, you can measure the net force of the object with this equation:

Fnet=Fweight- Fair. 
Two things can affect the force of air resistance
-speed
-surface area

When you first jump off, your speed begins to increase, this increases the force of air resistance and you acceleration starts to decrease, as does your Fnet(see equation with increasing Fair). This does not mean velocity starts to decrease as well, in fact it continues increasing. When the object reaches terminal velocity, it is at equilibrium.

But, if the object were say a person skydiving, and that person reached equilibrium and then opened the parachute, the acceleration and Force would both decrease because the Fair increases because the surface area increased. Then the skydiver will reach a second terminal velocity, however, it is going at a slower speed than the first.
*The force of air resistance is the same for both terminal velocities.