Introduction to Mechanics Centripetal Force

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1 Intoduction to Mechanics Centipetal Foce Lana heidan De Anza College Ma 9, 2016

2 Last time intoduced unifom cicula motion centipetal foce

3 Oveview using the idea of centipetal foce

4 Detemine (a) the astonaut s obital speed and (b) the Findingpeiod Centipetal of the obit. Foce Example del tutoial available in ebassign 3. In the Boh model of the hydogen atom, an electon moves in a cicula path aound a poton. The speed lem of the electon is appoximately m/s. Find toial available (a) in Enhanced the foce acting on the electon as it evolves in a Pagecicula 169, # obit 4 of BIOadius m and (b) the eo solution available centipetal in acceleation Q/C of the electon. ebassign 4. A cuve in a oad foms pat of a hoizontal cicle. As a ca goes aound it at constant speed 14.0 m/s, the total hoizontal foce on the dive has magnitude 130 N. What is the total hoizontal foce on the dive if the speed on the same cuve is 18.0 m/s instead? 5. In a cycloton (one type of paticle acceleato), a deuteon (of mass 2.00 u) eaches a final speed of 10.0% of the speed of light while moving in a cicula path of adius m. What magnitude of magnetic foce is equied to maintain the deuteon in a cicula path? 6. A ca initially taveling y a 7. A d 3 w t 8. C W o t m

5 Finding Centipetal Foce Example Page 169, # 4 F c = mv 2

6 Finding Centipetal Foce Example Page 169, # 4 F c = mv 2 m = F c = v N (14.0m/s) 2 = kg/m

7 Finding Centipetal Foce Example Page 169, # 4 F c = mv 2 m = F c = v N (14.0m/s) 2 = kg/m F c = m v 2 = ( kg/m)(18.0 m/s) 2 = 215 N

8 ngle u. Theefoe, if g is smalle, as it is on Mas, the speed v with which the oa. Feis Wheel Foces A Feis wheel is a ide you tend to see at fais and theme paks. Feis Wheel is wheel as shown n a vetical cicle of f 3.00 m/s. y the seat on the pess you answe in g. AM Top v R n bot n t v Figue 6.6a. Based on a Feis wheel o ay, you would expect path. Duing imilaly, the you ide the speed, v, is constant. Figue 6.6 (Example 6.5) (a) A child ides on a Feis whee bottom of the path. mg mg Bottom a b c

9 Feis Wheel Foces Quick Quiz You ae iding on a Feis wheel that is otating with constant speed. The ca in which you ae iding always maintains its coect upwad oientation; it does not invet. (i) What is the diection of the nomal foce on you fom the seat when you ae at the top of the wheel? (A) upwad (B) downwad (C) impossible to detemine 1 Page 153, eway & Jewett

10 Feis Wheel Foces Quick Quiz You ae iding on a Feis wheel that is otating with constant speed. The ca in which you ae iding always maintains its coect upwad oientation; it does not invet. (i) What is the diection of the nomal foce on you fom the seat when you ae at the top of the wheel? (A) upwad (B) downwad (C) impossible to detemine 1 Page 153, eway & Jewett

11 Feis Wheel Foces Quick Quiz You ae iding on a Feis wheel that is otating with constant speed. The ca in which you ae iding always maintains its coect upwad oientation; it does not invet. (ii) Fom the same choices, what is the diection of the net foce on you when you ae at the top of the wheel? (A) upwad (B) downwad (C) impossible to detemine 1 Page 153, eway & Jewett

12 Feis Wheel Foces Quick Quiz You ae iding on a Feis wheel that is otating with constant speed. The ca in which you ae iding always maintains its coect upwad oientation; it does not invet. (ii) Fom the same choices, what is the diection of the net foce on you when you ae at the top of the wheel? (A) upwad (B) downwad (C) impossible to detemine 1 Page 153, eway & Jewett

13 m m/s 2 dius banked at a fixed angle u. Theefoe, if g is smalle, as it is on Mas, the speed v with which the oadway s taveled that the speed ly is also smalle. v is popotional to the squae oot of g fo a oadway Finalize Equation (3) shows that the banking angle is independent of the mass of the vehicle negotiating the cuve. If a oe, g is the smalle, it is speed vacceleation with which thetheefoe, oadway caif ounds cuve at aas speed lesson thanmas, 13.4 m/s,the the centipetal deceases. the nomal foce, which is unchanged, is sufficient to cause two acceleations: the lowe centipetal acceleation and an acceleation of the ca down the inclined oadway. Consequently, an additional fiction foce paallel to the oadway and upwad is needed AM Top 6.5 to keepriding the cathe fomfeis slidingwheel down the bank (to the left in Fig.v 6.5). imilaly, a dive attempting to negotiate the cuve top net at a speed geate than 13.4 m/s has to depend on fiction to keep fom sliding up the bank (to the ight in Fig. 6.5). mass m ides on a Feis wheel as shown nbot H ATchild I F? moves Imagine this cicle same of oadway wee built on Mas in the futue to connect diffeent colony centes..6a. W The in a that vetical ntop it be speed taveled them/s. same speed? m atcould a constant of at 3.00 Feis Wheel Assume the speed, v, is constant. n < mg : F points down R el mine Answe theam focethe exeted by the seat ontop thefoce on Mas would mean that the ca is not pessed as tightly to the oadway. The educed gavitational bottom of the ide. Expess you answe inv component of the nomal foce towad the cente of the cicle. This smalle educed nomal foce esults in a smalle e weight of the child, component wouldmg. not be sufficient to povide the centipetal acceleation associated with the oiginal speed. The censhown the speed v. tipetal acceleation must be educed, which can be done by educing nbot N Mathematically, notice that Equation (3) shows that the speed v is popotional to the squae oot of g fo a oadway cleofof fixed adius banked at a fixed angle u. Theefoe, if g is smalle, as it is on Mas, the speed vnwith which the oadway v top ize Look caefully at Figue 6.6a. Based mg mg can be safely taveled is also smalle. nces you may have had on a Feis wheel o Bottom c small hills on a oadway, you would expect R a b he at the top of the path. imilaly, you hte Figue 6.6 (Example 6.5) (a) A child ides on a Feis wheel. ct to feel of the path. net bot we in heavie6.5at the bottom (b) The foces at the bottom of the path. AMacting on the child Top Example e bottom of the path and Riding the top,the the Feis no- Wheel top of the path. (c) The foces acting on the child at the v avitational foces on the child act in opposite childsum of mass m ides on a Feis TheAvecto of these two foces gives awheel as shown nbot inmagnitude Figue 6.6a. The child a vetical cicle of path at a constant speed. To yield net foce vecnstant that keeps themoves child in moving in a cicula adius 10.0 m at a constant speed of 3.00 m/s. e same magnitude, the nomal foce at the bottom must be geate than that at the top. R continued (A) Detemine the foce exeted by the seat on the v child at the bottom of the ide. Expess you answe in Based tems of the weight of the child, mg. mg mg n heel o Bottom OLUTION expect a Conceptualize Look caefully at Figue 6.6a. Based y, you on expeiences you may have had on a Feis wheel o Figue 6.6 b > mg : F points up ntop c v Bottom mg (Example 6.5) (a) A child ides on a Feis wheel. mg

14 EXAMPLE 6 8 Rounding a Cone A 1200-kg ca ounds a cone of adius = 45 m. peed, fiction, & centipetal foce example, Ex 6-8 If the coefficient of static fiction betwee what is the geatest speed the ca can have in the cone without skidding? A 1200-kg ca Pictue ounds the Poblem a cone of adius = 45 m. If the coefficient of static fiction between the ties and the oad is µ s = 0.82, what is the geatest speed the ca can have in the nomal foce, N! = NyN; and the foce of static fiction, f! s = m cone without skidding? s NxN. In the fist sketch we show a bid s-eye view of the ca as it moves along its cicula path. T diectly towad the obseve. Note that we have chosen the positive x diection to point towa the positive y axis to point vetically upwad. We also indicate the thee foces acting on the v ide view N Top view = 45 m W ketch fee body diagam fo ca. tategy In this system, the foce of static fiction povides the centipetal foce equied fo the ca to m foce of fiction is at ight angles to the ca s diection of motion; it is diected towad the cente in this case is static because the ca s ties ae olling without slipping always making stat the ca moves faste, moe centipetal foce (i.e., moe fiction) is equied. Thus, the geates maximum static fiction, f s = m s N. Hence, if we set m s N equal to the centipetal foce, ma cp = Hypothesis: The faste the ca goes, the lage the centipetal foce needed to stay in the tun. The centipetal foce will come fom static fiction, which cannot take a lage value than µ s n. It is a tight tun. Guess: 30 m/s.

15 peed, fiction, & centipetal foce example, Ex 6-8 = 45 m. If the coefficient of static fiction between the ties and the oad is m s = 0.82, e in the cone without skidding? A 1200-kg ca ounds a cone of adius = 45 m. If the coefficient of static fiction between the ties and the oad is µ s = 0.82, what is the geatest speed the ca can have in the cone without skidding? iew of the ca as it moves along its cicula path. The next sketch shows the ca moving e have chosen the positive x diection to point towad the cente of the cicula path, and ad. We also indicate the thee foces acting on the ca: gavity, the atic fiction, f! W! = -WyN = -mgyn; s = m s NxN. ketch: y ide view N f s x = 45 m W tategy: ovides the centipetal foce equied fo the ca to move in a cicula path. That is why the s diection of motion; it is diected towad the cente of the cicle. In addition, the fiction ae olling without slipping always making static contact with the gound. Finally, if e (i.e., moe fiction) is equied. Thus, the geatest speed fo the ca coesponds to the 2

16 peed, fiction, & centipetal foce example, Ex 6-8 = 45 m. If the coefficient of static fiction between the ties and the oad is m s = 0.82, e in the cone without skidding? A 1200-kg ca ounds a cone of adius = 45 m. If the coefficient of static fiction between the ties and the oad is µ s = 0.82, what is the geatest speed the ca can have in the cone without skidding? iew of the ca as it moves along its cicula path. The next sketch shows the ca moving e have chosen the positive x diection to point towad the cente of the cicula path, and ad. We also indicate the thee foces acting on the ca: gavity, the atic fiction, f! W! = -WyN = -mgyn; s = m s NxN. ketch: y ide view N f s x = 45 m W tategy: F net = ma cp and F cp = f s. y-diection: ovides the centipetal foce equied fo the ca to move in a cicula path. That is why the F s diection of motion; it is diected towad net,y = n mg = 0 the cente of the cicle. In addition, the fiction ae olling without slipping always making static contact n with= the gound. mg Finally, if e (i.e., moe fiction) is equied. Thus, the geatest speed fo the ca coesponds to the 2

17 peed, fiction, & centipetal foce example, Ex 6-8 A 1200-kg ca ounds a cone of adius = 45 m. If the coefficient of static fiction between the ties and the oad is µ s = 0.82, what is the geatest speed the ca can have in the cone without skidding? x-diection: F net,x = f s = mv 2 µ s n = mv 2 µ s (mg) = mv 2 v = µ s g = 19 m/s

the tun. The tendency you tun slightly to the dius and the centipet Cuved oadways ae often not flat.

A Banked Tun The steeply banked tack at the Talladega peedway in Alabama (left) h This is so that tangential a

Even of when thethee nomal is no foce solid oadway, on the ca howeve, can banking c help povide keep some fom o

18 the tun. The tendency you tun slightly to the dius and the centipet Cuved oadways ae often not flat. The ae often banked, that is skidding, and you can sloped at an angle to the hoizontal. A Banked Tun The steeply banked tack at the Talladega peedway in Alabama (left) h This is so that tangential a component path. Even of when thethee nomal is no foce solid oadway, on the ca howeve, can banking c help povide keep some fom o skidding all of thesideways. centipetal Banking foce. is beneficial in anothe way as w aiplane feel no sideways foce when the banking angle is just ight, so tu 1 Photo fom eason, Walke, some Physics. tains use hydaulic suspension systems to bank when ound

19 of tun. A Banked Tun A tun has a adius. What should the angle θ be so that a ca a cone taveling without at speed any vassistance can tun the fom cone fiction without between elyingthe onties fiction? and a taveling at 20.5 m/s in a tun of adius 85.0 m. lly enthe e, N y x W e a ht ust supply the needed centipetal foce. Thus, we find N by setting Then we use N in F x = ma x = mv 2 / to find the angle u.

20 of tun. A Banked Tun A tun has a adius. What should the angle θ be so that a ca a cone taveling without at speed any vassistance can tun the fom cone fiction without between elyingthe onties fiction? and a taveling at 20.5 m/s in a tun of adius 85.0 m. lly enthe e, N y x W e a ht ust supply the needed centipetal foce. Thus, we find N by setting Then Hint: we use conside N in F what x = the ma net x = foce mv 2 / vecto to find must the be angle in this u. case.

21 A Banked Tun ines the op- hout any assistance fom fiction between the ties and t 20.5 m/s in a tun of adius 85.0 m. y-diection (vetical): y N x F y,net = 0 n y W = 0 W he needed centipetal foce. Thus, we find N by setting N in F x = ma x = mv 2 / to find the angle u. a F y = N cos u - W = 0

22 A Banked Tun ines the op- hout any assistance fom fiction between the ties and t 20.5 m/s in a tun of adius 85.0 m. y-diection (vetical): N W y x F y,net = 0 n y W = 0 n cos θ = mg = W n = mg cos θ he needed centipetal foce. Thus, we find N by setting N in F x = ma x = mv 2 / to find the angle u. a F y = N cos u - W = 0

23 even if thee hout any assistance fom fiction between the ties and t 20.5 m/s in a tun of adius 85.0 m. y N x ines A the Banked op- Tun x-diection (hoizontal): F x,net = F c n x = mv 2 W he needed centipetal foce. Thus, we find N by setting N in F x = ma x = mv 2 / to find the angle u. a F y = N cos u - W = 0 N = W cos u = mg cos u

24 even if thee hout any assistance fom fiction between the ties and t 20.5 m/s in a tun of adius 85.0 m. N W he needed centipetal foce. Thus, we find N by setting N in F x = ma x = mv 2 / to find the angle u. a F y = N cos u - W = 0 N = W cos u = mg cos u y x ines A the Banked op- Tun x-diection (hoizontal): F x,net = F c n x = mv 2 n sin θ = mv 2 mg mv 2 sin θ = cos θ tan θ = v 2 g θ = tan 1 ( ) v 2 g

25 ummay applications of centipetal foce Homewok Walke Physics: Ch 6, onwad fom page 177. Questions: 13, 21; Poblems: 59, 61, 63, 73, 85, 107

PS113 Chapter 5 Dynamics of Uniform Circular Motion

PS113 Chapter 5 Dynamics of Uniform Circular Motion PS113 Chapte 5 Dynamics of Unifom Cicula Motion 1 Unifom cicula motion Unifom cicula motion is the motion of an object taveling at a constant (unifom) speed on a cicula path. The peiod T is the time equied