Describing Circular motion

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Brent Garrett
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1 Unifom Cicula Motion Descibing Cicula motion In ode to undestand cicula motion, we fist need to discuss how to subtact vectos. The easiest way to explain subtacting vectos is to descibe it as adding a negative vecto. Fo instance: When we add two vectos we add them head-to-tail A Fig 7.1 Hee in Fig 7.1 we have shown two fee vectos. We add them to find the esultant like this: Resultant Fig 7.2 A Since this is a vecto, not only do we need a magnitude to completely descibe it, we also need a diection. Fo the puposes of cicula motion, we will discuss diection a little late. When we want to subtact two vectos, think of it as adding a negative vecto. A negative vecto is defined as equal in magnitude to the oiginal, but opposite in diection. Fo example: A -A Fig 7.3 1

2 With this in mind, let s do the poblem A : i) A Fig 7.4a ii) _-A We add them Head-to-tail Fig 7.4b iii) -A Fig 7.4c Resultant Now that we undestand how to subtact vectos, conside cicula motion. An object s position elative to the cente of a cicle is given by the position vecto. As the object moves aound the cicle, the length of the position vecto does not change, but its diection does. To find the object s velocity, you need to find its displacement vecto ove a time inteval. The change in position, o the object s displacement, is epesented by. (emembe that means 2 1 ) v 1 Fig 7.5a Fig 7.5b 1 v

3 is the displacement duing the time inteval t, ove which time the object taveled. Recall that a moving object s aveage velocity is d, so fo an object in cicula motion, t = t motion.. The velocity vecto has the same diection as the displacement fo cicula At any point in the motion of the object as it tavels in the cicle, the instantaneous velocity vecto is tangent to the cicle. And emembe that at any point on a cicle, the tangent line is pependicula to the adius dawn to that point. So, in ou diagam, the velocity vectos ae pependicula to the position vectos at that given instant. To illustate this, imagine what would happen if you wee swinging a mass at the end of a ope in a cicle. Now imagine what the object s motion would be if the ope suddenly boke. The mass would tavel in a diection that was tangent to the oiginal cicula motion at the instant the ope boke. Of couse, hee on Eath, the mass would then follow a paabolic tajectoy of a pojectile. Centipetal acceleation The wod Centipetal is defined as: centipetal 1. Moving o diected towad a cente o axis. 2. Opeated by means of centipetal foce. 3. Physiology Tansmitting neve impulses towad the cental nevous system; affeent. 4. otany Developing o pogessing inwad towad the cente o axis, as in the head of a sunflowe, in which the oldest flowes ae nea the edge and the youngest flowes ae in the cente. 5. Tending o diected towad centalization: the centipetal effects of a homogeneous population. In you book, it is simply Cente-seeking. As you can see, the tem centipetal acceleation would be used to descibe acceleation whose diection is towad the cente of a cicle. Recall that acceleation is defined as the change in velocity pe unit time Fig 7.5a and Fig 7.5b, using the same method we used to detemine the change in displacement, we can find the change in velocity. v 1 v t. Refe to v 2 v Fig 7.6 3

4 Note that the diection of v in Fig 7.6 is towad the cente of the cicle. This einfoces the notion that the diection of the acceleation is towad the cente of the cicle. The nice thing that happens now between the tiangles in Fig 7.5b and Fig 7.6 is that they ae simila. If you emembe fom discussions in Geomety class about simila tiangles, you will emembe that in simila tiangles, the side lengths ae popotional. Using this fact, we can wite a popotion between the two tiangles as such: = v v This equation is not changed if we divide each side by t. t = v v t Since we said ealie that v = t and a c = v t equation to get: v = a c v, we can substitute these values into ou Finally, by solving fo a c, we aive at an equation fo centipetal acceleation. (use the coss-poduct) a c = v2 Remembe that the diection of the centipetal acceleation always points towad the cente of the cicula motion. The velocity in ou equation is the tangential instantaneous velocity at a given time. Howeve, in ode to measue the speed of an object moving in a cicle, we need to measue its peiod, T. Recall that peiod is the time it takes fo one complete oscillation o evolution. Duing this time, the object tavels a distance equivalent to the cicumfeence of the cicle, 2π. The object s speed, then, is epesented by v = 2π T. If we substitute this expession into ou equation fo centipetal acceleation, the following equation is: a c = 2π/T 2 4

5 Rewoking this equation we aive at: a c = 4π2 T 2 Since we have an acceleation, we must have a foce acting on the object. Recall fom ou discussion of Newton s Second Law that F = ma. This also applies to cases of centipetal acceleation as well. The foce that causes an object to tavel in a cicle is called the centipetal foce. In the case of a mass at the end of a sting, the centipetal foce is being applied by the tension in the sting. In the case of an obiting planet, gavity supplies the centipetal foce. When a ca goes aound a cuve, the inwad foce is the fictional foce of the oad on the ties. This foce, accoding to Newton s Second Law is as follows: Motion in a vetical cicle F c = ma c These equations descibe the motion of an object that is taveling at a constant speed in a cicle. Examples of this would be motion in a hoizontal cicle (like a game of tetheball), a sock in a spinning washing machine, o a penny sitting on a spinning ecod playe. All of these examples descibe unifom cicula motion. ut how does motion in a vetical cicle affect these equations? Conside a mass, m, at the end of a sting of length being whiled in a vetical cicle. Look at a fee-body diagam at a specific instant. + (adial) Fig 7.7 θ F t +t (tangential) F y mg F x In this figue, F t is the foce of tension in the ope, F y is the vetical component of the foce of weight, and F x is the hoizontal component of the foce of weight. In Figue 7.7, you will notice that the foce of gavity is acting on the mass. ecause we ae including the mass of the object, the speed is not constant aound the cicle. The object speeds up on the downside and slows down on the up side. This means the 5

6 tangential acceleation, a t 0 acceleation is a c = v2. v t 0. Still, the magnitude of the adial (centipetal) If we conside any point in the motion of the object, we can wite Newton s Second Law fo the adial diection as follows: And fo the tangential diection: F t F y = Fc = mv2 F x = ma t whee a t is the acceleation in the tangential diection So what happens at the bottom and top of the cicle? At the bottom of the cicle, the velocity of the object is at a maximum v max. Fom any point beyond the bottom of the cicle, the object begins slowing down. The object slows down until it eaches the top of the cicle. At the top of the cicle, the velocity of the object is at a minimum (v min ). At any point beyond the top of the cicle, the object begins speeding up. This elationship continues as the object continues to spin in along this path. How does this affect the foces acting in this system? At the top and bottom of the cicle, we only conside the adial diection because the acceleation only has a adial component at those points. ottom Top v min Fig 7.8a Fig 7.8b F t2 F w F t1 v max F w In figue 7.8a, we look at what happens at the bottom of the cicle whee the velocity of the object is at a maximum. The foce of tension that the ope must apply in ode to keep the mass, m, in cicula motion is a combination of the centipetal foce, F c and the foce of weight, F w. F t1 = F c + F w 6

7 If we ewite the equation to solve fo the centipetal foce we get: F c = m(v max ) 2 = F t1 F w In figue 7.8b, we look at what happens at the top of the cicle whee the velocity of the object is at a minimum. The foce of tension that the ope must apply in ode to keep the mass, m, in cicula motion is a combination of the centipetal foce, F c and the foce of weight, F w. F t2 = F c F w If we ewite the equation to solve fo the centipetal foce we get: F c = m(v min ) 2 = F t2 + F w At the top of the cicle, something cuious happens if the object is spinning just fast enough to keep the ope taut but slow enough so that the ope does not have to apply any foce. O in othe wods, whee F t2 = 0. So what is the minimum speed fo which the sting emains taut? F c = m(v min ) 2 F c = m(v min ) 2 = F t2 + F w = F w m(v min ) 2 = mg m(v min ) 2 = mg v min 2 = g v min = g This minimum velocity is called citical velocity. This is the velocity at which the object must tavel in ode to stay in cicula motion. If you have eve swung a bucket of wate ove you head o ode a looping olle coaste, you can see why knowing how to calculate this value could be of geat impotance. 7

8 Example poblems: 1. A 25.0 kg child moves with a speed of 1.93 m/s when sitting 12.5m fom the cente of a mey-go-ound. Calculate a) the centipetal acceleation and b) the centipetal foce. 2. A N ca taveling at 50.0 km/h ounds a cuve of adius 2.00 x 10 2 m. Find a) the centipetal foce b) the centipetal acceleation of the ca, and c) the minimum coefficient of fiction between the ties and the oad so that the ca can ound the cuve safely. 3. A looping olle coaste ide has a adius of cuvatue of 7.50m. At what minimum speed must the coaste be taveling at the top of the cuve so that the passenges will not fall out? 8

Uniform Circular Motion

Uniform Circular Motion Unifom Cicula Motion Intoduction Ealie we defined acceleation as being the change in velocity with time: a = v t Until now we have only talked about changes in the magnitude of the acceleation: the speeding