Motion along curved path *

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1 OpenStax-CNX module: m Motion along cuved path * Sunil Kuma Singh This wok is poduced by OpenStax-CNX and licensed unde the Ceative Commons Attibution License 2.0 We all expeience motion along a cuved path in ou daily life. The motion is not exactly a cicula motion. Howeve, we can think of cuved path as a sequence of cicula motions of dieent adii. As such, motion of vehicles like that of ca, tuck etc, on a cuved oad can be analyzed in tems of the dynamics of cicula motion. Clealy, analysis is done fo the cicula segment with the smallest adius as it epesents the maximum cuvatue. It must be noted that "cuvatue" and "adius of cuvatue" ae invese to each othe. Motion along a cuved path Figue 1 One common expeience, in this espect, is the expeience of a ca dive, which is negotiating a shap tun. If a peson is sitting in the middle of the back seat, she/he holds on the xed pop to keep the postue steady and move along with the motion of the ca. If the peson is close to the fathe side (fom the cente of motion) of the ca, then she/he leans to the side of the ca to become pat of the motion of the ca. In eithe of the two situations, the equiement of centipetal foce fo cicula motion is fullled. The bottom of the body is in contact with the ca and moves with it, wheeas the uppe pat of the body is not. When she/he leans on the side of the ca away fom the cente of motion, the side of the ca applies nomal foce, which meets the equiement of centipetal foce. Finally, once the equiement of centipetal foce is met, the complete body is in motion with the ca. * Vesion 1.5: Aug 4, :30 am

2 OpenStax-CNX module: m Motion along a cuved path Figue 2: The peson pesses by side of the ca. The diection in which the body esponds to cuved motion is easy to nd by just thinking what would be the natual diection of motion of the fee pat of the body. In the example discussed above, the body seeks to move staight, but the lowe pat in contact with ca moves along cuved path having side way component of motion (towads cente). The esult is that the uppe pat is away fom the cente of cuvatue of the cuved path. In ode to keep the body upight an extenal foce in the adial diection is equied to be applied on the body. 1 Negotiating a level cuve Let us concentate on what is done to tun a ca to negotiate a shap tun. We ealize that the dive of a ca simply guides the steeing of the ca to move it along the tun. Intuitively, we think that the ca engine must be esponsible fo meeting the equiement of centipetal foce. This is ight. Howeve, engine does not diectly apply foce to meet the equiement of centipetal foce. It is actually the fiction, esulting fom the motion caused by engine, which acts towads the cente of cicula path and meets the equiement of centipetal foce. The body of the ca tends to move staight in accodance with its natual tendency? Now, as wheel is made to move side way (by the change in diection), the wheel has the tendency to have elative motion with espect to the oad in the diection away fom the cente of path. In tun, oad applies fiction, which is diected towads the cente.

3 OpenStax-CNX module: m Negotiating a level cuve Figue 3: The fiction meets the equiement of centipetal foce. Impotant to note hee is that we ae not consideing fiction in the fowad o actual diection of motion, but pependicula to actual diection (side way). Thee is no motion in the side way diection, if thee is no side way skidding of the ca. In that case, the fiction is static fiction ( f s ) and has not exceeded maximum o limiting fiction ( F s ). Thus, f s F s f s Thee is no motion in vetical diection. Hence, Combining two equations, we have : µ s N N = mg f s µ s mg whee m is the combined mass of the ca and the passenges. In the limiting situation, when the ca is about to skid away, the fiction foce is equal to the maximum static fiction, meeting the equiement of centipetal foce equied fo cicula motion, mv2 = µ s mg v = ( µ s g )

4 OpenStax-CNX module: m We note following points about the condition as put on the speed of the ca : Thee is a limiting o maximum speed of ca to ensue that the ca moves along cuved path without skidding. If the limiting condition with egad to velocity is not met ( v ( µ s g ) ), then the ca will skid away. The limiting condition is independent of the mass of the ca. If fiction between ties and the oad is moe, then we can negotiate a cuved path with highe speed and vice-vesa. This explains why we dive slow on slippey oad. Smalle the adius of cuvatue, smalle is the limiting speed. This explains why shape tun (smalle adius of cuvatue) is negotiated with smalle speed. 2 Banking of oads Thee ae thee additional aspects of negotiating a cuve. Fist, what if, we want to negotiate cuve at a highe speed. Second, how to make diving safe without attacting limiting conditions as ties may have been attened (whose gooves have attened), o fiction may decease due to any othe easons like ain o mud. Thid, we want to avoid sideway fiction to polong life of the ties. The answe to these lie in banking of the cuved oad. Fo banking, one side of the oad is elevated fom hoizontal like an incline o wedge. In this case, the component of nomal foce in hoizontal diection povides the centipetal foce as equied fo the motion along the cuved path. On the othe hand, component of nomal foce in vetical diection balances the weight of the vehicle. It is clea that magnitude of nomal eaction between oad and vehicle is geate than the weight of the vehicle.

5 OpenStax-CNX module: m Banking of oads Figue 4: The hoizontal component of nomal foce meets the equiement of centipetal foce. Hee, we compute the elation between the angle of banking (which is equal to the angle of incline) fo a given speed and adius of cuvatue as : and Ncosθ = mg Taking atio, Nsinθ = mv2 tanθ = v2 g v = ( gtanθ ) This expession epesents the speed at which the vehicle does not skid (up o down) along the banked oad fo the given angle of inclination (θ). It means that centipetal foce is equal to the esultant of the system of foces acting on the vehicle. Impotantly, thee is no fiction involved in this consideation fo the cicula motion of the vehicle. Example 1 Poblem : An aicaft hoves ove a city awaiting cleanace to land. The aicaft cicles with its wings banked at an angle tan 1 ( 0.2 ) at a speed of 200 m/s. Find the adius of the loop.

6 OpenStax-CNX module: m Solution : The aicaft is banked at an angle with hoizontal. Since aicaft is executing unifom cicula motion, a net foce on the aicaft should act nomal to its body. The component of this nomal foce in the adial diection meets the equiement of centipetal foce, wheeas vetical component balances the weight of aicaft. Thus, this situation is analogous to the banking of oad. tanθ = v2 g = v2 gtanθ = x 0.2 = m 3 Role of fiction in banking The moot question is whethe banking of oad achieves the objectives of banking? Can we negotiate the cuve with highe speed than when the oad is not banked? In fact, the expession of speed as deived in ealie section gives the angle of banking fo a paticula speed. It is the speed fo which the component of nomal towads the cente of cicle matches the equiement of centipetal foce. If speed is less than that specied by the expession, then vehicle will skid "down" (slip o slide) acoss the incline as thee is net foce along the incline of the bank. This educes the adius of cuvatue i.e. "" is educed - such that the elation of banking is held tue : v = ( gtanθ ) Banking of oad Figue 5: The vehicle has tendency to skid down. In eality, howeve, the inteacting sufaces ae not smooth. We can see that if fiction, acting "up" acoss the bank, is sucient to hold the vehicle fom sliding down, then vehicle will move along the cicula path without skidding "down". What would happen if the vehicle exceeds the specied speed fo a given angle of banking? Clealy, the equiement of centipetal foce exceeds the component of nomal foce in the adial diection. As such the vehicle will have tendency to skid "up" acoss the bank. Again fiction pevents skidding "up" of the vehicle acoss the bank. This time, howeve, the fiction acts downwad acoss the bank as shown in the gue.

7 OpenStax-CNX module: m Banking of oad Figue 6: The vehicle has tendency to skid up. In the nutshell, we see that banking helps to pevent skidding "up" acoss the bank due to the equiement of centipetal foce. The banking enables component of nomal foce in the hoizontal diection to povide fo the equiement of centipetal foce up to a cetain limiting (maximum) speed. Simultaneously, the banking induces a tendency fo the vehicle to skid "down" acoss the bank. On the othe hand, fiction pevents skidding "down" as well as skidding "up" acoss the bank. This is possible as fiction changes diection opposite to the tendency of skidding eithe "up" o "down" acoss the bank. The state of fiction is summaized hee : 1: v = 0; f S = mgsinθ, acting up acoss the bank 2: v = gtanθ; f S = 0 3: v < gtanθ; f S > 0, acting up acoss the bank 4: v > gtanθ; f S > 0, acting down acoss the bank Fiction, theefoe, changes its diection depending upon whethe the vehicle has tendency to skid "down" o "up" acoss the bank. Stating fom zeo speed, we can chaacteize fiction in following segments (i) fiction is equal to the component of weight along the bank," mgsinθ ", when vehicle is stationay (ii) fiction deceases as the speed inceases (iii) fiction becomes zeo as speed equals " gtanθ " (iv) fiction changes diection as speed becomes geate than " gtanθ " (v) fiction inceases till the fiction is equal to limiting fiction as speed futhe inceases and (vi) fiction becomes equal to kinetic fiction when skidding takes place. 3.1 Skidding down the bank In this case, the velocity of the vehicle is less than theshold speed " gtanθ ". Fiction acts "up" acoss the bank. Thee ae thee foces acting on the vehicle (i) its weight "mg" (ii) nomal foce (N) due to the bank suface and (iii) static fiction " f s ", acting up the bank. The fee body diagam is as shown hee.

8 OpenStax-CNX module: m Banking of oad Figue 7: The speed is less than the theshold speed. Fx Nsinθ f S cosθ = mv2 Fy Ncosθ + f S sinθ = mg 3.2 Skidding up the bank In this case, the velocity of the vehicle is geate than theshold speed. Fiction acts "down" acoss the bank. Thee ae thee foces acting on the vehicle (i) its weight "mg" (ii) nomal foce (N) due to the bank suface and (iii) static fiction " f s ", acting down the bank. The fee body diagam is as shown hee.

9 OpenStax-CNX module: m Banking of oad Figue 8: The speed is geate than the theshold speed. Fx Nsinθ + f S cosθ = mv2 Fy Ncosθ f S sinθ = mg 4 Maximum speed along the banked oad In pevious section, we discussed vaious aspects of banking. In this section, we seek to nd the maximum speed with which a banked cuve can be negotiated. We have seen that banking, while peventing upwad skidding, ceates situation in which the vehicle can skid downwad at lowe speed. The design of bank, theefoe, needs to conside both these aspects. Actually, oads ae banked with a small angle of inclination only. It is impotant as geate angle will induce tendency fo the vehicle to ovetun. Fo small inclination of the bank, the tendency of the vehicle to slide down is uled out as fiction between tyes and oad is usually much geate to pevent downwad skidding acoss the oad. In pactice, it is the skidding "up" acoss the oad that is the pime concen as theshold speed limit can be beached easily. The banking supplements the povision of centipetal foce, which is othewise povided by the fiction on a at oad. As such, banking can be seen as a mechanism eithe (i) to incease the theshold speed limit o (ii) as a safety mechanism to cove the isk involved due to any eventuality like attening of tyes o wet oads etc. In fact, it is the latte concen that pevails.

10 OpenStax-CNX module: m In the following paagaph, we set out to detemine the maximum speed with which a banked oad can be negotiated. It is obvious that maximum speed coesponds to limiting fiction that acts in the downwad diection as shown in the gue. Banking of oads Figue 9: The hoizontal component of nomal foce and fiction togethe meet the equiement of centipetal foce. Foce analysis in the vetical diection : Foce analysis in the hoizontal diection : Taking atio of two equations, we have : Ncosθ µ s Nsinθ = mg N ( cosθ µ s sinθ ) = mg Nsinθ + µ s Ncosθ = mv2 N ( sinθ + µ s cosθ ) = mv2 g ( sinθ + µscosθ ) ( cosθ µ ssinθ ) = v2 ( sinθ + µscosθ ) ( cosθ µ ssinθ ) ( tanθ + µs ) ( 1 µ stanθ ) } v 2 = g v = { g

11 OpenStax-CNX module: m Bending by a cyclist We have seen that a cyclist bends towads the cente in ode to move along a cicula path. Like in the case of ca, he could have depended on the fiction between ties and the oad. But then he would be limited by the speed. Futhe, fiction may not be sucient as contact suface is small. We can also see "bending" of cyclist at geate speed as an altenative to banking used fo fou wheeled vehicles, which can not be bent. The cyclist inceases speed without skidding by leaning towads the cente of cicula path. The sole objective of bending hee is to change the diection and magnitude of nomal foce such that hoizontal component of the nomal foce povides fo the centipetal foce, wheeas vetical component balances the "cycle and cyclist" body system. Banking of oads Figue 10: The hoizontal component of nomal foce meets the equiement of centipetal foce. Ncosθ = mg Taking atio, Nsinθ = mv2 tanθ = v2 g v = ( gtanθ )

Chapter 5. really hard to start the object moving and then, once it starts moving, you don t have to push as hard to keep it moving.

Chapter 5. really hard to start the object moving and then, once it starts moving, you don t have to push as hard to keep it moving. Chapte 5 Fiction When an object is in motion it is usually in contact with a viscous mateial (wate o ai) o some othe suface. So fa, we have assumed that moving objects don t inteact with thei suoundings