Unit 5 Summary

Work and Power

work = a transfer of power
work = (force)(distance)   --> measured in joules (J)
*but remember that force and distance must be parallel for work to be done

For example, if a man was walking down the street holding a dog, no work would be done on the dog because the man's distance is moving forward and the dog's gravity is downward.
... But if the man was picking up the dog from the ground, there would be work on the dog, because the man is moving upward thus making the force and the distance parallel.

There is work being done, because the man's weight is 600N and the stairs have a vertical height of 4m
--> the force and distance are parallel
*the slanted height of the stairs DOES NOT matter because it isn't parallel with the force


Example problem: There are two people who weigh 500N. One man walked up the stairs, and the other man took the elevator. Which person did gravity do more work on?
-->

*There is an exception to the force and distance parallel rule.

Work can still be done if something is slanted, because gravity is still parallel to the y-component, so it doesn't have to be completely parallel to the actual distance (velocity in this case).





power = how quickly work is done
power = work/time   --> measured in watts
*If you halve the time, you double the work

Work and Kinetic Energy

kinetic energy - energy of movement

kinetic energy = .5mv^2   --> measured in joules (J)

work = ∆ kinetic energy

*if you have mass & velocity, you have the potential to do work

Say an 8g bike was moving at 10m/s and then 10 seconds later was moving at 20m/s. What was the 

∆ kinetic energy?

KE initial = .5mv^2                 KE final = .5mv^2                  

                  = .5(8)(10)^2                          = .5(8)(20)^2
                  = 4(100)                                  = 4(400)
                  = 400 J                                    = 1,600J

∆ kinetic energy = KE final - KE initial
                             = 1,600 J - 400J
                             = 1,200J  <--------- ∆ KE

work = ∆ KE so since there was 4x as much KE in KE new than in KE original, we know there was 4x as much work in KE new than in KE original.

*remember when you're solving for work that kg IS NOT a force, so you must convert the kg into N using w=mg because weight IS a force.
Go two decimal places to the left when converting from kg --> N
Go two decimal places to the right when converting from N --> kg

Potential Energy

potential energy = mgh --> (mass)(gravity)(height)
*movement IS NOT necessary to have potential energy
*the amount of potential energy is affected by the height (energy of position), so if something has height it has potential energy.

PE      = ∆KE        = work
(mgh) = (.5mv^2) = (F)(d)

The potential energy = the kinetic energy
         (potential)                     (actual)





*The Law of Conservation of Energy says that, "the total amount of energy in a system remains constant, or is conserved. Energy can't be created nor destroyed, but it can change form." Just like in this problem PE = KE so the energy went from potential energy to kinetic energy.

PE = mgh 
*if you double the height, you double the PE
*if you double the mass, you double the PE
*if you double the height & mass, you quadruple the PE
*gravity always remains constant .98 or 10

Machines

These are three different machines.
left = ramp
middle = pulley
right = lever

The purpose of machines is to decrease the amount of force needed, thus making the job easier.

The work put in = the work that comes out
work in = work out 
  (F)(d)  =  (F)(d)

In terms of work and energy, how do airbags keep you safe?
Airbags keep you safe, because you go from moving to not moving no matter how you're stopped, because ∆KE is the same with or without the airbag because (KE = .5mv^2) and
∆KE = KE final - KE initial. Since the ∆KE is the same, the work is also constant because
(∆KE = work). The airbag increases the distance to stop, and since work has to be the same, the force will decrease with the airbag because (work = Fd), thus having a smaller force.
(small force = small injury = safer)
work = Fd <-- no airbag
work = fD <-- airbag

In terms of work and energy, why do gymnasts use mats?
Gymnasts use mats because they keep them safer. They will go from moving to not moving no matter how they're stopping because ∆KE is the same with or without a mat because (KE = .5mv^2) and
(∆KE = KE final - KE initial). Since the ∆KE is the same, the work is also constant because
(∆KE = work). The mats increase the distance to stop, and since the work has to be the same, the force will decrease with the mats because (work = Fd), thus having a smaller force.
(small force = small injury = safer)
work = Fd <-- no mat
work = fD <-- mat

Unit 4 Summary

Rotational Inertia/Angular Momentum

Inertia is an object's resistance to change. 

Rotational Inertia is the property of an object to resist changes in the a spin. For example, an object with a smaller amount of rotational inertia is easier to spin than an object with a larger rotational inertia.

If you remember the Conservation of Linear Momentum that states (total momentum before = total momentum after), you will remember that momentum is ALWAYS conserved. So...

The Conservation of Angular Momentum is the law that states
(angular momentum before = angular momentum after).

Angular momentum requires two things

1) rotational inertia

2) rotational velocity

So... angular momentum = (rotational inertia)(rotational velocity)

(rotational inertia)(rotational velocity) = (rotational inertia)(rotational velocity)

This is the before = after

For example 

The ice skater on the left has her arms out. This puts her arms' mass farther from her axis of rotation, giving her a larger rotational inertia and thus a smaller rotation velocity. The ice skater on the right has her arms pulled in. This puts her arms' mass closer to her axis of rotation, giving her a smaller rotational inertia and thus a larger rotational velocity. This is why ice skaters spin faster when their arms are pulled in to their chests.

Rotational & Tangential Velocities 

Same rotational-different tangential --> Train wheels are designed to keep the train on the track by having the same rotational speed, but different tangential speeds. They're designed to have narrow edges on the outside and a wider wheel on the inside. This causes the wheels to have the same rotational velocities (moving the same distance over the same period of time) because they make the same amount of rotations. When the wider wheel ends upon the far side of the track then that side will be going faster and will cause the train to curve back towards the middle (self-correcting).

Different rotational-same tangential --> Gears are designed to have the same tangential speeds. Even when a smaller gear and a larger gear are connected, they are going the same speed. Although, the smaller wheel isn't making as many rotations as the larger gear, thus giving the two gears different rotational velocities (the amount of rotations made during a period of time).

Torques

A torque is a force acting over a perpendicular distance (lever arm), and it is caused by rotation. A torque happens when an object's center of gravity goes outside of it's base of support. 

This object's center of gravity is outside it's base of support, so a torque is caused. This object will be easy to knock over.

*An object will be more likely to have a torque if its base is narrow or if it has a high center of gravity, and will be less likely to have a torque if its base is wide or if it has a low center of gravity.

This object's center of gravity is inside it's base of support, so a torque is NOT caused. This object will be hard to knock over. 

*An object will be more likely to have a torque if its base is narrow or if it has a high center of gravity, and will be less likely to have a torque if its base is wide or if it has a low center of gravity.

This system is balanced, because the torques are equal. 

Torque = (force)(lever arm)

Counter-clockwise torque = clockwise torque

(F)(lever arm) = (F)(lever arm)


*It is important to remember that...
1) force does NOT = force
2) lever arm does NOT = lever arm
3) torque = torque 

Centripetal Force 

Centripetal force is the force that makes a body follow a curved path.
*Centrifugal force (the center fleeing force) is NOT a real force. 

This ball is connected to a string attached to a ceiling. When the ball is pushed, it will continue moving in a circular path due to centripetal force, which causes the ball to rotate around its axis of rotation (where the string connects to the ceiling)

String = F tension

Ball = F weight
*The centripetal force is in the inward direction, because it will NEVER be outward.

Example: In a washing machine, there are little holes on the inside metal to let the water drain out of the clothes. The clothes have a centripetal force in addition to their tangential velocity. The water only has a tangential velocity, because it is small enough to fit through the holes. It will keep moving in a straight line through the holes due to its tangential velocity. If there were no holes, then the water would have a centripetal force and stay inside the washer.

Center of Gravity 

Example: Wrestlers bend their legs when wrestling, because it lowers their center of gravity. This makes it harder to get their center of gravity outside of their base of support, thus making it harder to create a torque. It essentially makes them more stable.

Example: A handy-man has wrench on a stuck bolt and can't seem to get the bolt loose. How can he make his job easier? He should get a longer wrench, because that will create a longer lever arm (the distance from the force applied to the axis of rotation). Now there is a larger lever arm AND a larger force, so there is a larger torque. Torque = (F)(lever arm)

Example: If a steel ball and a tape roll were to race down a ramp, the steel ball would win. The steel ball's weight is evenly distributed throughout the object, while the tape roll has a hole in the center so its mass is farther from its axis of rotation, giving it a higher rotational inertia and thus a smaller rotational velocity. The steel ball will have a smaller rotational inertia and thus a larger rotational velocity.

Example: If a water bottle filled with frozen water and a water bottle filled with regular water were to race down a ramp, the regular water bottle would win. In the frozen water bottle, everything is rotating when the bottle rotates, because it is all frozen together. In the regular water bottle, the bottle is rotating around the water, because the water for the most part stays in place.