Done with Unit Four and Ready for More!

Unit 4 was all about circular motion (rotation), including: center of mass, rotational and tangential velocity, torque, rotational inertia, centripetal force and conservation of angular momentum.

Center of Mass

The center of mass of an object is the average position of its mass and the center of gravity is the specific point upon which gravity acts. These two points are not always  the same. The center of mass and center of gravity are pertinent to the topic of rotation because their location can affect how easily an object can rotate.

Whether or not an object will fall over or not can be determined by the location of its center of gravity over its base of support. If the center of gravity falls inside of an object's base of support, it is stable. However, if the center of gravity falls outside of the base of support, the object is unstable.

For example, look at these images of the leaning Tower of Pisa.

The base of support of an object is its base. For the leaning Tower of Pisa, the base of support is the circumference of the bottom of the tower. However, for people, our base of support is based (pun semi-intended) on how wide apart we have our feet planted. The wider our base of support, the harder it is to push us over. This is why coaches often tell players to stand with their legs bent, shoulder width apart. The reason we bend our legs is because the closer our center of gravity is to our base of support, the farther we will have to rotate to get the center our center of gravity outside of our base of support. 

At the end of this video by some classmates, this concept is illustrated by some members of the wrestling team.

Rotational and Tangential Velocity

Tangential speed is the same linear speed we've talked about in past units. In terms of circular motion, tangential speed is the linear speed of something moving along a circular path. The direction of the linear motion is tangent to the circumference of the circle. Just like usual, this is measured in meters per second (m/s) or kilometers per hour (km/h).

Rotational speed is the speed at which an object rotates over an amount of time. This is measure in rotations per minute (rpm).

Watch this video of  kids on a merry go round, if the girl in the orange shirt and the boy in the striped shirt have the same rotational velocity, what does this mean about their tangential velocities?


The girl in the orange shirt has a faster tangential velocity than the boy in the striped shirt because she must cover a farther linear distance than he does in the same amount of time.

Conversely, gears in a system work by having the same tangential velocity and a different rotational velocity. This can be observed in the video below:





Torque

Torque is what causes rotation. It is calculated using the formula force x lever arm. Lever arm can be defined as the distance between the axis of rotation and where the force is applied. It is perpendicular to the force. If a large rotation occurs, it is because there is a large torque. Large torque can be caused by a large force, a large lever arm or both.

When an object is balanced, there are two torques: clockwise and counter-clockwise. The two are equidistant from the center of gravity.

An example of how torque affects daily life is when you try to open a door with a push handle. If you push on the door closer to the hinges, the axis of rotation, you need more force to open the door because the lever arm is small. However, pushing at the opposite end of the door makes it easy to open because of the distance between where your hands are pushing on the door and the hinges--thus giving you a larger torque.

This video gives a good explanation of torque and calculations.


Rotational Inertia

Rotational inertia is the property of an object to resist changes in spin. Since we know that with inertia more mass means more inertia, we also know that the distribution of mass is important when it comes to what makes an object inclined to spin or remain still. A great example of distribution of mass and rotational inertia is in figure skating when the skaters tuck in their arms to spin faster. Watch as this skater's speed increases when she tucks her arms and legs in.



Conservation of Angular Momentum

This is a concept that goes hand in hand with rotational inertia and is similar to concepts we've discussed in the past. Just like the conservation of linear momentum, this means that momentum before = momentum after. Angular momentum is calculated by multiplying rotational inertia x rotational velocity.

Angular momentum before = angular momentum after.

RI x RV before = RI x RV after.

Let's put this in terms of the video we just watched. Before she tucks in her arms and legs her rotational inertia is very large, but her rotational velocity is much smaller. Therefore, to conserve her angular momentum after she tucks in her arms and legs, her rotational inertia must be very small and her rotational velocity must be much larger.

That looks like this:

RI x RV = RI x RV

If you're still having trouble with the concept, this video has a quick explanation and lots of good (and easy to re-create!) demonstrations.



Centripetal Force

Centripetal force is the force that makes an object curve. Centripetal force is the reason you stay inside of a roller coaster that goes in loop-di-loops and the reason clothes stay inside a top-loading washing machine during the spin cycle.

Here's an explanation of why water leaves the barrel of a washing machine, but clothes stay in:

Centripetal force is the center-seeking force hat causes an object to curve. The water droplets are acted upon by the centripetal force and the water droplets are not being acted upon by any force. They just continue to move straight forward, straight through the holes, which we know happens because of the property of inertia.

Centrifugal force does not exist and is not the reason that you hit the car door when you round corners or why the water droplets come out of the basin. Those simply happen because as a person/object in motion without an outside for acting upon you/it forward motion continues until an outside force is met.

These are some cool experiment done to show centripetal force.

We found the mass of a meter stick without using a scale!

The goal of this lab was to find the mass of a meter stick with our new knowledge of torque, a meter stick and a 100g lead weight.

First we did a demo with a meter stick unbalanced on the edge of a table with a torque. In order to make sure that we understood how to label the parts of torque (force and lever arm) we made diagrams.

The first diagram just showed the unbalanced meter stick.

The next diagram had a meter stick balanced, showing the relationship between center of gravity and the edge of the table. The center of gravity can never be over the base of support so there is only one lever arm in this situation. It is important to remember that the lever arm only stretches between the axis of rotation and where the force is applied.

When the 100g mass is added to the meter stick, the lever arm in the counterclockwise torque extends from the axis of rotation to the end of the meter stick only because that is where the force from the 100g lead mass is applied. However, on the clockwise side, the lever arm stretches from the axis of rotation to the center of gravity of the entire meter stick because that is where the force of gravity is applied. On the meter stick, the center of gravity is the 50cm mark and will always be the 50cm mark.

Plan:

Our plan to use our knowledge from unit 4 was as follows.

1. Find the center of gravity of the counter-clockwise side (with the mass) by finding the centimeter mark on which it is balanced on the edge of the table.
2. Measure the lever arms on each side.
3. Use the lever arms and the force of gravity (9.8N) to find the clockwise and counter-clockwise torques.
4. Use w=mg to find the mass.

It took us a couple of tries to figure out exactly what numbers to plug in, but when we figured it out, this is what it looked like:

• Counter-clockwise lever arm: the distance from the center of gravity to the 100g lead weight
• 24.8cm
• Clockwise lever arm: center of gravity - the counter-clockwise lever arm
• 25.2cm
• Force on the counter-clockwise side: force of gravity / 100g
• 0.98N
• Force on the clockwise side: unknown

Our equation to set the counter-clockwise and clockwise torques equal to each other ended up looking like this:

Then we multiplied it out to find:

 

 



We then divided each side by 25.2 in order to isolate the force and found that:

 

We know that force is equal to mass x gravity so we divided by the force of gravity to find the mass:

However since we were looking for the mass of the meter stick in grams, we had to divide the mass in kilograms by 1000 and found that:

 







 

Torque and Center of Mass and Center of Gravity, Oh My!

Torque is force multiplied by the perpendicular distance from the axis of rotation (also called a lever arm.) Torque causes rotation and the size of the torque is directly proportional to the size of the lever arm, meaning that a long lever arm means a big torque. In the video below, I find that Ultimate Physics Tutor does a great job of reinforcing the basics while still expanding on the concept in his second explanation using vectors.

 

Rotational inertia is the property of an object to resist changes in spin. Since we know that with inertia more mass means more inertia, we also know that the distribution of mass is important when it comes to what makes an object inclined to spin or remain still. A great example of distribution of mass and rotational inertia is in figure skating when the skaters tuck in their arms to spin faster.



Just like linear momentum, angular momentum is conserved. This means that the skater's angular momentum before she tucked in her arms and legs is the same as her angular momentum after. Angular momentum is calculated by multiplying rotational inertia by rotational velocity. Since we know her angular momentum is conserved, we also know that before she tucked her rotational inertia must have been significantly larger than her rotational velocity and that after she tucked her rotational inertia must have been significantly smaller than her rotational velocity.

This quick video has nice explanations and good examples of ways this concept can be demonstrated at home.

Recap of Unit Three, Yippee!

Unit 3 was based in Newton's Third Law and its surrounding concepts and applications to real life. We covered Newton's Third Law, action-reaction pairs, how tug-of-war really works, forces in perpendicular directions, how tides work, momentum, impulse and the Law of Conservation of Momentum.

Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. This is exemplified all around us in action-reaction pairs.


In this image, there are two sets of action-reaction pairs.

 In the first, the book pushes down on the table and, with the same amount of force, the table pushes up on the book. Note that in this action-reaction pair, the subject switches from book to table, but the verb stays the same, push.

The second action-reaction pair is a little bit different. In this pair, gravity pulls the table down and the table pulls the earth up. When writing action-reaction pairs involving gravity, the subjects will always be gravity and the earth, but the verb will stay the same.

How does tug-of-war really work?

Now that Newton's Third Law has been covered, it's time to face a harsh truth about tug-of-war. The secret to winning does not lie in the strength of one team versus the other, but in physics.

Watch this video, and consider how Newton's Third Law might be applied.

There are three action-reaction pairs at work in a game of tug-of-war.

1. Team A pulls Team B to the left, Team B pulls Team A to the right.
2. Team A pushes the ground forward, the ground pushes Team A backward.
3. Team B pushes the ground forward, the ground pushed Team B backward.

Now, because we know that for every action there is an equal and opposite reaction, we can also know that the two teams are pulling with the same amount of force. Therefore, no matter the teams are stacked, one team pulling harder than the other is not how the match is won. In fact, it is the relationship between the team and the ground that determines the winner. In the video, the team on the left won because they were pushing on the ground with more force than the team on the right was.



This same concept can be applied to a horse and cart. It may seem that because of Newton's Third Law, a horse and a cart would pull each other with equal force and therefore would remain at rest. However, the tug-of-war explanation can also be used to prove that logic to be uninformed.

Like in tug-of-war, there are three action-reaction pairs: between the horse and the cart, between the horse and the ground and between the cart and the ground. The horse pulls the cart with and equal and opposite amount of force that the cart pulls the horse with. This is because of Newton's Third Law. However, the horse pushes on the ground and the ground pushes on the horse with a greater amount of force than the cart pushes on the ground and the ground pushes on the cart with. Therefore, the cart is pulled in the horse's direction.

Forces in Perpendicular Directions

Momentum

The definition of momentum is inertia in motion. In other words, momentum is the product of an object's mass and velocity. The unit for momentum is kgm/s^2 (kilogram meters per second, squared).

The equation for momentum (symbol p) is p=mv.

NOTE: At rest, there is no momentum, which can just be written as 0kgm/s^2.

Impulse
Impulse (symbol J) is defined as force multiplied by the amount of time it takes to be applied. In other words, the equation is J=f x change in time. Impulse is measured in Newton seconds (Ns).

Momentum/ Impulse Relationship

So, now that impulse and momentum have been defined, it is now the time to explore the relationship between the two. Momentum is equal to mass times velocity and change in momentum is equal to an object's final momentum minus its initial momentum.

NOTE: The change in momentum will always be the same whether or not the object comes to a stop slowly or quickly.

Impulse is equal to change in momentum.

Let's take this further by looking at how this relationship applies to a real life situation. Why is it that airbags keep us safe?


First, remember that the car will go from moving to not moving no matter how it is stopped. Therefore impulse/change in momentum is constant. A small force therefore must mean a large change in time and a large force means a small change in time. Airbags protect the dummy because they increase the amount of time it takes for the dummy to feel the impact of the collision, therefore the force acting on the dummy is decreased. Small force = less injury.

The Law of Conservation of Momentum

The Law of Conservation of Momentum states that the momentums before and after an action are equal, so it is conserved. For instance, say that there are two carts on a track, one is at rest and the other is in motion. The law states that if these two carts collide, connect and continue down the track as a unit, the momentum of the system would be the same before and after the collision.

As an equation, this looks like p total before= p total after.

Let's use a to name the moving cart and b to label the cart at rest.

We can use the equation: mass of a times velocity of a + mass of b time velocity of b  = mass of a + mass of b (velocity of unit ab).

Now let's imagine that cart a has a mass of 2kg and is moving left at 5m/s toward cart b which has a mass of 3kg and a velocity of 0m/s. When they collide, connect and continue, is the momentum conserved?

p total before=p total after

p of a + p of b before=p of a + p of b after

mava+mbvb=ma+b(vab)

2(5)+3(0)=2+3(vab)

10+0=5(vab)

2m/s=vab

Now that we know the velocity of the unit, we can use p=mv to solve for the momentum before and after.

Before:

2kg(5m/s)+3kg(0m/s)=p

10kgm/s^2=p

After:





5kg(2m/s)=p

10kgm/s^2=p

So we've found that the momentum is conserved. 

But...is momentum conserved when to pool balls hit and then move away from each other at an angle?

Yes!

Gravity and Tides

The Universal Gravitational Law states that F=(G)(m1)(m2)/d^2, because of this we know that force is proportional to 1/distance squared. In other words, a short distance has a strong force and a long distance has a weak force.

Let's think about this in relationship to the Earth. The side of the Earth closer to the Moon experiences the Moon's gravitational force more than the side that is farther away from the Moon. The net force acting on each side of the Earth is what causes the tides. Say that the on the side closer to the Moon, the Moon's gravitational force exerts 150N and on the other side, only 50N. (note: these numbers are grossly exaggerated just to detail the concept) Additionally, the gravitational force from the center of the Earth is 100N. To find the net force on each side, the Earth's gravitational force is subtracted from the moon's force on each side. Therefore, the net force on the close side is 50N and the net force on the far side is -50N. The positive force indicates that the close side is being pulled toward the Moon and the negative force indicates that the far side is being pulled away from the moon. It is this imbalance that causes the tides. If the forces weren't opposite, tides wouldn't exist. Also, if tides were determined by gravitational pull instead of the relationship between force and distance, tides would be caused by the Sun, not the Moon.

This video explains tides really well:

Tides Explanation

The Universal Gravitational Law states that F=(G)(m1)(m2)/d^2, because of this we know that force is proportional to 1/distance squared. In other words, a short distance has a strong force and a long distance has a weak force.






Let's think about this in relationship to the Earth. The side of the Earth closer to the Moon experiences the Moon's gravitational force more than the side that is farther away from the Moon. The net force acting on each side of the Earth is what causes the tides. Say that the on the side closer to the Moon, the Moon's gravitational force exerts 150N and on the other side, only 50N. (note: these numbers are grossly exaggerated just to detail the concept) Additionally, the gravitational force from the center of the Earth is 100N. To find the net force on each side, the Earth's gravitational force is subtracted from the moon's force on each side. Therefore, the net force on the close side is 50N and the net force on the far side is -50N. The positive force indicates that the close side is being pulled toward the Moon and the negative force indicates that the far side is being pulled away from the moon. It is this imbalance that causes the tides. If the forces weren't opposite, tides wouldn't exist. Also, if tides were determined by gravitational pull instead of the relationship between force and distance, tides would be caused by the Sun, not the Moon.




This video explains tides really well:










In each day, there are two high tides and two low tides due to planetary revolution. Six hours pass between each high tide and each low tide, also meaning that from high tide to high tide is 12 hours and between low tide to low tide is 12 hours.

Unit Three Resource

So far, we've covered in class that Newton's third law states that for every action there is an equal and opposite reaction. Meaning, that as I type and rest my hands on my laptop, my laptop is pushing up on my hands with the same amount of force.

That being said, does the Earth pull on the Moon with the same force that the Moon pulls on the Earth?

This video by Veritasium gives a little explanation and debunks a common misconception.



By now, hopefully it is clearer why Newton's third law is possible.

Now: why is it that no one wins the tug of war in this video?
(note: embedding was disabled for this video, but I really thought it was helpful)

No Win Tug of War

Newton's third law also rings true in this case. The two scientists pulled on the rope with the same force, however since there was no interaction between their feet and the ground, the scientists pulled each other together instead of one dominating the other.

See an example of successful tug of war:



The team on the left won not because they were stronger, but because the action-reaction force pair between their feet and the ground was greater than that on the right.

That's a Wrap with Unit Two, Whew!

Alas, this is the conclusion of Unit 2 in Physics class. This past unit we covered: Newton's second law, free fall and falling through the air.

Newton's Second Law

Newton's second law of motion states that acceleration is directly proportional to force and inversely proportional to mass. The formula for Newton's second law is a=F/m which can also be written as a=F*1/m.

Newton's Second Law Lab

The lab we did in class on Newton's second law was centered around the question how does acceleration depend on force and mass? The purpose was to discover how the acceleration of a system is related to its mass and to its force. The system was a cart, with a hanging weight attached, on a track.

In Experiment A, we kept the force on the system constant, but added mass. Since, as Newton's second law states, acceleration and mass are inversely proportional the mass added to the system caused the acceleration to decrease. The force on the system was the force of weight from the hanger. The force on the system stayed constant because the hanging weight was unchanged.

When graphing the data from Experiment A, we used the formula y=mx+b. In order to translate that into physics, we had to be mindful of what was kept constant. The force remained constant throughout the experiment, so it became the slope. Then, I wrote out the conversion, which can be done in two way.

Conversion Type 1:

 

 

 

Since we know that F is our slope, we can look at y=mx as F=mx. Then, we can think about which of our formulas that resembles, which is F=ma from Unit 1. Now we know where to plug in force, mass and acceleration when graphing.

Conversion Type 2:

Here, we just line up the formula for Newton's second law with y=mx. We know how to line it up because we've already identified force and the slope.

In Experiment B, the mass of the system remained constant, but the force was increased. This was achieved by moving 100kg weight from the cart to the hanger, one trial at a time. For this experiment, the slope was equal to the mass.

Free Fall

Free fall is when an object falls due to the acceleration of gravity only. Remember that the acceleration due to gravity is 9.8m/s^2.

Things Falling Straight Down

The information above can be used to find out the height from which an object falls from. For instance, if a ball falls down a cliff and it takes 20 seconds for the ball to hit the ground, how high was the cliff. First of all, the acceleration due to gravity can be rounded up to 10 m/s^2 because this is a real life situation. Also, since this is a vertical situation, the equation used is d=1/2gt^2.

d=1/2gt^2

The first step is to plug in all given information into the equation.
 d=1/2(10)(20)^2

Next, simplify.

d=1/2(10)(40)
d=1/2(400)
d=200

So, d=400m. That means that the cliff was 400 meters high.

The ball's velocity can be found using v=gt.

v=gt
v=(10)(20)
v=200m/s

The ball's velocity was 200 meters per second.


This is a video created by some classmates that includes two more practice problems.

Throwing Things Straight Up


This is a video I made with two of my classmates about things being thrown straight up.

Falling at an Angle

With objects falling at an angle, it is vital to remember two types of special right triangles: 3,4,5 and 10, 10, square root of 10. Also, remember that the square root of two is equal to 1.41. In these situations, the hypotenuse of the triangle will equal the actual velocity. 

This video by some classmates gives an example problem that might be helpful.


Projectile Motion

In projectile motion, the same free fall formulas can be used, so here's the chart again.

An example of projectile motion is when someone throws a football.

REMEMBER: vertical height controls the distance in the air. Always.



This video by my classmates has a great breakdown explanation of projectile motion.

Falling with Air Resistance

A real life example of falling with air resistance is skydiving. Watch this video, and pay attention to the skydiver's velocity as he falls to the earth. The music is distracting, but there is nice display of his velocity starting at 1:00.





This is my favorite video by my classmates, explaining what happens when air resistance is included when falling, like when you have a parachute.

Here's another video about skydiving and physics.



Wrap-up

There are a few things that are vital to Unit 2. First are the formulas from the handy dandy chart I made.

Second, the acceleration due to gravity is equal to 9.8m/s^2 (in real life situations, otherwise just round to 10).

Third, vertical height controls time in the air.

Fourth, velocity is always constant and downward when falling through the air with air resistance.