Circular Motion. x-y coordinate systems. Other coordinates... PHY circular-motion - J. Hedberg

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Cicula Motion PHY 207 - cicula-motion - J. Hedbeg - 2017 x-y coodinate systems Fo many situations, an x-y coodinate system is a geat idea. Hee is a map on Manhattan.

Thus, when we gie diections in NY, its usually something like "1 block up, 2 blocks oe", which in moe pope math speak would be something like 1 unit in the y, and 2 units in the x diection.

Thus, they each ely on thei own coodinate system, which is somewhat like what we'll use fo descibing cicula motion.

1 Cicula Motion PHY cicula-motion - J. Hedbeg x-y coodinate systems Fo many situations, an x-y coodinate system is a geat idea. Hee is a map on Manhattan. The steets ae laid out in a ectangula gid (fo the most pat - dang it Boadway). Thus, when we gie diections in NY, its usually something like "1 block up, 2 blocks oe", which in moe pope math speak would be something like 1 unit in the y, and 2 units in the x diection. Othe coodinates... Fo othes, not so much In the case of a dat boad, o a baseball field, descibing the position of something in x-y coodinates wouldn't make much sense. Thus, they each ely on thei own coodinate system, which is somewhat like what we'll use fo descibing cicula motion. If we walk aound a cicle, we can descibe ou position with espect to the cente by two aiables: θ and. We could also use x and y, but we can see how just using an angle and a adial distance is easie. This is the essence of what will be known as the adial coodinates system (o pola). It's ey useful when does not change, o if θ is a constant. It's easie to wok with than the standad x-y catesian coodinates when object ae undegoing cicula motion because the does not change. Thus, we only hae to woy about one aiable, θ. If we used x and y, then we would hae to keep tack of two aiables. Moe wok. Less fun. Quick Question 1 a) d dt b) dθ dt If a paticle is moing in a cicle that is situated in the x-y plane at a constant speed, which of the following quantities will be zeo? c) dx dt d) dy dt Page 1

2 What is a Radian? PHY cicula-motion - J. Hedbeg "The length of an ac of a unit cicle is numeically equal to the measuement in adians of the angle that it subtends" Cicle Math If θ is measued in adians, the distance aound the cued path of the cicle can be found by: θ = s This makes sense consideing: θ full cicle s 2π = = = 2π ad In this setup, s is known as the ac length. It's distance aound the cicumfeence of the cicle that the blue line takes. Fo the whole cicle, this alue is equal to the cicumfeence, o 2π. Peiod A sometimes moe useful measue of the obit of an object is its peiod, T. The peiod is defined as the time it takes to complete one eolution aound. Fo example: 1. The peiod of the moon aound the eath: days = s 1 2. The peiod of a LP ecod is 1.8 s (33 ot pe minute) 3 The peiod can be calculated by consideing the speed and the cicumfeence of the obit: distance taeled 2π 2π = = T = time T Now let's conside an object moing with the same speed along the cicula path. At any gien point, the elocity ecto is both tangent to the cicle and has the same length at all positions. The total time that it takes fo this object to go aound the cicle once the peiod, T. Thus, we can say the speed (the magnitude of the elocity) is distance taeled = = time it took 2π T Page 2

PHY 207 - cicula-motion - J. Hedbeg - 2017 Quick Question 2 The adius of Columbus Cicle is 0.025 miles (1/40th of a mile). If you want to die aound it in 5 seconds (1/720 hous).

3 PHY cicula-motion - J. Hedbeg Quick Question 2 The adius of Columbus Cicle is miles (1/40th of a mile). If you want to die aound it in 5 seconds (1/720 hous). How fast should you be going? a) 12 π mph b) 24 π mph c) 36 π mph d) 72 π mph e) 144 π mph Example Poblem #1: Find the speed of the moon as it obits aound the eath. The moon is on aaage 385,000 km fom the eath. It take 27.3 days to complete one obit. ` Unifom Cicula Motion: If we can say that the speed of an object undegoing cicula motion is not changing, then we'll call that type of motion: Unifom Cicula Motion. sun satun neptune pluto Some examples: planets going aound the sun. (mostly cicula although not quite) A ecod playe. hoses on the caousel (while the ide is not stating o stopping.) Quick Question 3 A coin sits on a otating tuntable. At the instant shown in the figue, which aow is closest to the diection of the coin s elocity? Let's find Δ: Δ = 0 Gaphical subtaction would look like this: 0 Page 3

4 PHY cicula-motion - J. Hedbeg Quick Question 4 Which ecto best epesents 0? A B C D 0 0 Now we can see whee the acceleation ecto is pointing. It's pointing towad the cente of the cicle. With a little geomety, we could show that the magnitude of the acceleation ecto will be gien by:. 0 Thus, an object undegoing unifom cicula motion has a 'centipetal acceleation' popotional to the speed squaed and the inesely popotional to the distance fom the cente. Sim a = 2 Quick Question 5 Rank the following places along the oad in the ode of lagest centipetal acceleation. (assume the ca has a constant speed) a) a 1 > a 2 > a 3 > a 4 b) a 2 > a 1 > a 4 > a 3 c) a 1 = a 2 = a 3 > a 4 d) a 2 > a 3 > a 1 > a 4 e) a 4 > a 2 > a 3 > a 1 Page 4

Centipetal Foce PHY 207 - cicula-motion - J.

centipetal foce. This is not a new foce, it's meely a classification of the peious foces we'e aleady dealt with. A ball is spun aound on a sting.

5 Centipetal Foce PHY cicula-motion - J. Hedbeg Newton's second law still holds in the case of cicula motion F = ma = m 2 We'll can any foce (tension, fiction, gaity, etc which is always pointed towads the cente of a cicula path a centipetal foce. This is not a new foce, it's meely a classification of the peious foces we'e aleady dealt with. A ball is spun aound on a sting. The tension foce is always diected towads the cente of the cicula path. A ca taeling aound a cue is acted on by the static fiction foce. Eath aound stays in obit due to the adially diected foce of gaity between the eath and the sun. Play/Pause Let's spin a ball on a sting fa away fom the eath so thee is no gaity acting on the ball. Thus the only foce is pesent is the tension fom the ope. Example Poblem #2: F = T Putting this into Newton's second law, we'll hae: 2 T = m a cent = m A sting will beak if the tension in it is geate than 500 N. How fast can I swing a 1 kg ball using 2 metes of this sting without beaking it? I'm swinging it in a hoizontal plane (disgegad gaity fo now). Page 5

PHY 207 - cicula-motion - J. Hedbeg - 2017 Quick Question 6 A coin sits on a otating tuntable. At the instant shown in the figue, which aow gies the diection of the fictional foce on the coin?

6 PHY cicula-motion - J. Hedbeg Quick Question 6 A coin sits on a otating tuntable. At the instant shown in the figue, which aow gies the diection of the fictional foce on the coin? Play/Pause The static fiction between the metal coin and the wood will be stong enough to keep the coin fom slipping, poided the of the wooden plate doesn't get too lage. Hee, the only foce acting the adial diection is the. Thus: f s 2 F = f s = m Quick Question 7 A B C Hee ae thee identical blocks on a otating disk. At low otation speeds they all stay in place on the platfom due to static fiction. As the otation speed inceases, which block will the fist to fly off? a) Block A b) Block B c) Block C d) They all fly off at the same time Example Poblem #3: What s the maximum speed my 200 kg go-kat can go aound a cue with adius 10m without sliding out of contol? (Assume ubbe ties on dy concete.) So fa... Objects taeling aound in a cicle, een with constant speed, hae a centipetal acceleation, gien by: a cent = 2 This is a ecto, and always pointed towads the cente of the cicle aound which the object is taeling. Also pointing towads the cente of the cicle is a centipetal foce. This could be tension, fiction, gaity, etc... F cent = m a cent The peiod of an object undegoing unifom cicula motion is gien by: T = 2π Page 6

PHY 207 - cicula-motion - J. Hedbeg - 2017 Quick Question 8 Hee is a ball being swung in a etical cicle. Among the points listed, whee is the tension in the sting the geatest?

7 PHY cicula-motion - J. Hedbeg Quick Question 8 Hee is a ball being swung in a etical cicle. Among the points listed, whee is the tension in the sting the geatest? a) point A b) point B c) point C d) point D e) it's the same eeywhee Quick Question 9 What is an expession fo the nomal foce acting on the mass ( m) if the tay is being acceleated in the diection of +y with a magnitude of a? a) F N = mg b) F N = m(g a) c) F N = m(g + a) d) = m(a/g) F N +y m a Nomal Foces Swinging Scales Play/Pause At the position shown, what is the minimum speed the box can hae while still touching the tay? Example Poblem #4: A space station is designed in the shape of a lage, hollow ing that is unifomly otating. The oute diamete of the station is 300 m. With what peiod must the station otate so that a peson sitting on the oute wall expeiences a moon like gaitational feel". Page 7

Quick Question 10 Which of the ectos best shows the net acceleation of the ball at the location shown? PHY 207 - cicula-motion - J.

8 Quick Question 10 Which of the ectos best shows the net acceleation of the ball at the location shown? PHY cicula-motion - J. Hedbeg At the position labeled A, what is the minimum speed the ball can hae while still keeping some tension on the ope? (i.e how fast to ensue tension cannot be zeo) Page 8

DYNAMICS OF UNIFORM CIRCULAR MOTION

DYNAMICS OF UNIFORM CIRCULAR MOTION Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object