We started the unit by learning about tangential velocity and rotational velocity. My class found out that rotational velocity is about the number of rotations over a time. When talking about rotational inertia we discovered that if, for example, two kids are playing on a merry-go-round regardless of their position their rotational velocity is the same! So, does that mean their tangential velocity is the same? No! Tangential velocity depends on how far away the body is from the axis of rotation. If the body is far away from the axis of rotation it needs to cover a greater distance per rotation therefore it has a greater tangential velocity, the opposite is true for an object close to the axis of rotation.
tangential speed ~ radial distance x rotational speed
Rotational inertia:
Rotational inertia is the same as linear inertia but for rotating objects. Just like objects moving in a linear path, rotating objects do not want to stop doing what they are already doing. Rewriting Newton's first law we can say:
An object rotating about an axis tends to remain rotating around an axis unless interfered with by some external force and an object not rotating wants to remain that way unless an outside force is applied.
Rotational inertia depends on concentration of mass and how far is it from the axis of rotation. The furthest away the concentration of mass is from the center the greater it's rotational inertia is. I struggled to convince myself that the greater the rotational inertia the hardest it is to start rotating, I always wanted to say that it would start rotating sooner than an object with a small rotational inertia. The secret to understand this concept is always to remind yourself that the inertia is laziness and the lazier the object is the more reluctant it would be to start rotating!
The video above shows that because the hoop has it's concentration of mass away from the axis and the disk is solid (both have the same mass). The hoop has more rotational inertia and therefore is more reluctant to start rotating and therefore loses the race.
Torque:
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| http://microship.com/resources/resourcepix/cgdb-1.jpg microship.com |
Center of Mass/Weight and Stability:
Center of mass is the average position of all the mass that makes up the object. If the object is symmetrical the center of mass is at the geometrical center. Center of mass is called center of gravity when weight is involved. To balance an object one needs to hold it on the center of mass and that will support the entire object and thus it will be balanced. Stability has everything to do with center of gravity. If the center of balance of an object is directly above it's base it will have balance if it is not it is unbalanced. The concept of center of mass and stability explains why you lean forward when carrying a heavy backpack, you want to keep your center of mass over your feet so you can keep your balance. The boy in the picture above is not leaning forward and that is why he feels unbalanced. If he can shift his center of mass so that is over his feet maybe he can make in time for class!
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| http://staff.fcps.net/sms-news/archive/2006-07/vol1issue1/6th/backpack.JPG staff.fcbps.net Dylan Kimmel |
Centripetal force is a force directed towards the center. If you have a can and a sting that your swing above your head you can notice the effect of centrifugal force. Even though you may think that there is a force pulling the can out there is only the force of tension of the string pulling it to the center. When talking about centripetal force we used the spin cycle of the of a washing machine as an example. The fact that the clothes are drier after the spin cycle is explained because the water and clothes have the same tangential velocity the water can go through the wholes in the side because there is no external force stopping them from continuing moving in the straight line they want to keep following since it has inertia. The sides of the washing machine provide a force of friction on the clothes that for that reason are pushed inward (centripetal force). It is possible to think that there is a centrifugal force on the clothes on the washing machine since they stay up against the washing machine walls but centrifugal force does not exist! The clothes seemed to be pushed outward because of the combination of the velocity of the clothes that is tangential to the circle with the friction force from the walls of the washing machine.
Conservation of Angular Momentum:
Just like linear momentum angular momentum is also conserved. The fact that angular momentum is conserved allows us to establish a relationship between it's two component: rotational inertia and rotational velocity. Angular momentum is rotational velocity times rotational inertia, since angular momentum is conserved they have a inversely proportional relation. The inversely proportional relation means, in other words, that if rotational inertia increases rotational velocity will decrease and the other way around. The ballerina on the video bellow is an example of conservation of angular momentum. The girl extends her leg and arms and that increases her rotational inertia (because her mass is farther away from the axis of rotation) and therefore decreases her rotational velocity since angular momentum is conserved. The ballerina probably has no idea but the fact that she decreases her velocity after every turn allows her to be able to put her foot flat on the ground and regain balance to do another pirouette.
This unit was probably one of the trickiest on the sense that all the concepts seemed to overlap. When trying to answer homework questions it was sometimes difficult to understand which concept to use. After some practice it became easier to distinct what concept the question was actually asking about. I struggled understanding how the train wheels work, what helped me to understand this was reading about on the book again. After reading about it I figured out that it was all about different tangential velocities and that they cause the train to self correct along it's way.
