Calculate coefficient of friction top top a vehicle tire.Calculate best speed and angle of a automobile on a turn.

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Any force or combination of forces can reason a centripetal or radial acceleration. Simply a couple of examples room the anxiety in the rope ~ above a tether ball, the pressure of Earth’s heaviness on the Moon, friction in between roller skates and also a rink floor, a banked roadway’s pressure on a car, and forces ~ above the tube of a turn centrifuge.

Any net force causing uniform circular motion is dubbed a centripetal force. The direction the a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. Follow to Newton’s 2nd law of motion, net pressure is mass time acceleration: network F = ma. For uniform one motion, the acceleration is the centripetal acceleration—ac. Thus, the size of centripetal pressure Fc is Fc = mac.

By utilizing the expressions because that centripetal acceleration ac native a_c=fracv^2r;a_c=romega^2\, we get two expressions because that the centripetal pressure Fc in regards to mass, velocity, angular velocity, and also radius of curvature: extF_c=mfracv^2r; extF_c=mromega^2\.

You may use whichever expression because that centripetal pressure is more convenient. Centripetal force Fc is always perpendicular come the path and also pointing to the facility of curvature, because ac is perpendicular to the velocity and pointing to the center of curvature.

Note that if you solve the first expression for r, you get displaystyler=fracmv^2 extF_c\.

This indicates that for a provided mass and also velocity, a huge centripetal force reasons a little radius that curvature—that is, a tight curve.

Figure 1. The frictional pressure supplies the centripetal force and also is numerically same to it. Centripetal pressure is perpendicular come velocity and also causes uniform circular motion. The bigger the Fc, the smaller sized the radius that curvature r and the cheater the curve. The second curve has the very same v, yet a larger Fc produces a smaller sized r′.

Example 1. What Coefficient of Friction Do automobile Tires need on a level Curve?

Calculate the centripetal force exerted ~ above a 900 kg vehicle that negotiates a 500 m radius curve in ~ 25.0 m/s.Assuming one unbanked curve, find the minimum revolution coefficient of friction, in between the tires and the road, static friction being the factor that keeps the car from slipping (see number 2).Strategy and also Solution for component 1

We know that extF_c=fracmv^2r\. Thus,

displaystyle extF_c=fracmv^2r=fracleft(900 ext kg ight)left(25.0 ext m/s ight)^2left(500 ext m ight)=1125 ext N\.

Strategy for part 2

Figure 2 shows the forces acting ~ above the car on an unbanked (level ground) curve. Friction is to the left, maintaining the automobile from slipping, and also because that is the just horizontal pressure acting ~ above the car, the friction is the centripetal pressure in this case. We recognize that the maximum revolution friction (at i m sorry the tires roll however do not slip) is μsN, wherein μs is the static coefficient of friction and N is the common force. The typical force amounts to the car’s weight on level ground, so the N=mg. Therefore the centripetal pressure in this instance is

Fc = fμsN = μsmg.

Now we have actually a relationship between centripetal force and also the coefficient of friction. Using the an initial expression because that Fc from the equation

egincases extF_c=mfracv^2r\ extF_c=mromega^2endcases, ext mfracv^2r=mu_smg\

We resolve this for μs, noting the mass cancels, and also obtain


Solution for part 2

Substituting the knowns,

displaystylemu_s=fracleft(25.0 ext m/s ight)^2left(500 ext m ight)left(9.80 ext m/s^2 ight)=0.13\.

(Because coefficients of friction are approximate, the prize is offered to only two digits.)


We could also solve part 1 utilizing the an initial expression inegincases extF_c=mfracv^2r\ extF_c=mromega^2endcases\ because m,v, and also r are given. The coefficient that friction uncovered in part 2 is lot smaller 보다 is generally found between tires and roads. The automobile will quiet negotiate the curve if the coefficient is higher than 0.13, due to the fact that static friction is a responsive force, being able to assume a value much less than yet no more than μsN. A greater coefficient would certainly also permit the vehicle to negotiate the curve in ~ a higher speed, but if the coefficient of friction is less, the safe rate would be much less than 25 m/s. Note that massive cancels, implying the in this example, the does not issue how heavily loaded the vehicle is come negotiate the turn. Mass cancels because friction is assumed proportional come the regular force, which consequently is proportional to mass. If the surface ar of the road were banked, the normal force would be less as will be debated below.

Figure 2. This automobile on level ground is relocating away and turning to the left. The centripetal pressure causing the automobile to rotate in a circular route is due to friction in between the tires and also the road. A minimum coefficient that friction is needed, or the automobile will relocate in a larger-radius curve and leave the roadway.

Let united state now consider banked curves, where the steep of the road helps you negotiate the curve. See number 3. The better the edge θ, the much faster you have the right to take the curve. Race tracks for bikes and also cars, because that example, often have steeply banked curves. In an “ideally banked curve,” the angle θ is such the you can negotiate the curve at a specific speed there is no the help of friction between the tires and also the road. We will certainly derive one expression for θ for an ideally banked curve and consider an instance related come it.

For best banking, the net exterior force equates to the horizontal centripetal force in the absence of friction. The materials of the normal pressure N in the horizontal and also vertical directions have to equal the centripetal force and the load of the car, respectively. In cases in which forces are no parallel, the is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions.

Figure 3 shows a cost-free body diagram for a car on a frictionless banked curve. If the angle θ is best for the speed and radius, climate the net external force will equal the crucial centripetal force. The only two exterior forces acting on the vehicle are its load w and the normal force of the road N. (A frictionless surface have the right to only exert a force perpendicular come the surface—that is, a normal force.) these two forces must include to provide a net exterior force that is horizontal toward the facility of curvature and has size mv2/r. Since this is the an important force and it is horizontal, we use a coordinate mechanism with vertical and also horizontal axes. Only the normal pressure has a horizontal component, and so this have to equal the centripetal force—that is,

Nsin heta=fracmv^2r\.

Because the auto does not leave the surface ar of the road, the network vertical pressure must be zero, an interpretation that the vertical contents of the two exterior forces have to be equal in magnitude and also opposite in direction. Native the figure, we watch that the upright component the the normal force is N cos θ, and the just other vertical pressure is the car’s weight. These must be same in magnitude; thus, N cos θ = mg.

Now us can integrate the last two equations to remove N and get an expression for θ, together desired. Solving the second equation for N=fracmgcos heta\ , and substituting this into the very first yields

displaystyleeginarray\mgfracsin hetacos heta=fracmv^2r\mg anleft( heta ight)=fracmv^2r\ an heta=fracv^2rgendarray\

Taking the train station tangent gives

heta= an^-1left(fracv^2rg ight)\ (ideally banked curve, no friction).

This expression can be interpreted by considering just how θ relies on v and also r. A big θ will be obtained for a large v and also a tiny r. That is, roads should be steeply banked for high speeds and also sharp curves. Friction helps, because it allows you to take the curve at better or reduced speed 보다 if the curve is frictionless. Note that θ walk not depend on the fixed of the vehicle.

Figure 3. The automobile on this banked curve is moving away and transforming to the left.

Example 2. What Is the ideal Speed to take a Steeply Banked tight Curve?

Curves on part test tracks and also race, such together the Daytona global Speedway in Florida, are an extremely steeply banked. This banking, v the assist of tires friction and really stable automobile configurations, enables the curve to be taken at really high speed. Come illustrate, calculation the rate at i beg your pardon a 100 m radius curve banked at 65.0° must be propelled if the road is frictionless.


We an initial note the all state in the expression because that the best angle that a banked curve other than for speed space known; thus, we need only rearrange the so the speed appears on the left-hand side and also then substitute known quantities.


Starting with

an heta=fracv^2rg\, we get v = (rg tan θ)1/2.

Noting that tan 65.0º = 2.14, we obtain

eginarray\v=left^1/2\ ext =45.8endarray\


This is just around 165 km/h, constant with a very steeply banked and also rather spicy curve. Tires friction enables a car to take the curve at significantly greater speeds.

Calculations similar to those in the preceding instances can it is in performed because that a host of interesting situations in i beg your pardon centripetal force is involved—a number of these are presented in this chapter’s Problems and Exercises.

Take-Home Experiment

Ask a friend or relative to swing a golf club or a tennis racquet. Take suitable measurements to estimate the centripetal acceleration of the end of the society or racquet. Friend may choose to execute this in slow motion.

PhET Explorations: Gravity and Orbits

Move the sun, earth, moon and space station come see just how it affects your gravitational forces and orbital paths. Visualize the sizes and also distances between different heavenly bodies, and also turn off gravity to check out what would take place without it!


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Section Summary

Centripetal pressure Fc is any kind of force leading to uniform circular motion. It is a “center-seeking” pressure that always points towards the center of rotation. The is perpendicular to linear velocity v and has actually magnitude Fc = mac, which can also be to express as

egincases extF_c=mfracv^2r\ extor\ extF_c=mromega^2endcases\

Conceptual Questions

If you great to alleviate the tension (which is pertained to centripetal force) top top high-speed tires, would you usage large- or small-diameter tires? Explain.Define centripetal force. Can any type of pressure (for example, tension, gravitational force, friction, and also so on) be a centripetal force? deserve to any combination of pressures be a centripetal force?If centripetal force is directed toward the center, why execute you feel that you space ‘thrown’ far from the center as a automobile goes approximately a curve? Explain.Race automobile drivers routinely cut corners as shown in number 7. Describe how this permits the curve to be taken at the best speed.

Figure 7. Two paths approximately a race track curve are shown. Race vehicle drivers will take the inside course (called cut the corner) whenever possible because it enables them to take it the curve in ~ the highest possible speed.

A variety of amusement parks have actually rides that make upright loops prefer the one presented in number 8. For safety, the cars room attached to the rails in such a method that lock cannot autumn off. If the automobile goes over the optimal at just the best speed, heaviness alone will certainly supply the centripetal force. What other force acts and what is that direction if: (a) The vehicle goes end the height at faster than this speed? (b) The vehicle goes over the peak at slower 보다 this speed?

Figure 8. Amusement rides through a upright loop are an instance of a kind of bent motion.

What is the direction the the force exerted by the auto on the passenger together the automobile goes over the peak of the amusement journey pictured in number 8 under the following circumstances: (a) The vehicle goes over the top at together a speed that the gravitational pressure is the only force acting? (b) The auto goes end the top quicker than this speed? (c) The automobile goes over the top slower 보다 this speed?As a skater develops a circle, what pressure is responsible because that making she turn? usage a cost-free body chart in her answer.Suppose a child is riding ~ above a merry-go-round at a distance around halfway between its center and also edge. She has a having lunch box relaxing on wax paper, so that there is very tiny friction in between it and the merry-go-round. Which path displayed in figure 9 will the lunch box take when she allows go? The having lunch box pipeline a follow in the dust ~ above the merry-go-round. Is the trail straight, bent to the left, or curved to the right? describe your answer.

Figure 9. A boy riding ~ above a merry-go-round releases her lunch crate at suggest P. This is a see from above the clockwise rotation. Assuming it slides v negligible friction, will certainly it follow path A, B, or C, as viewed from Earth’s structure of reference? What will certainly be the shape of the route it pipeline in the dust ~ above the merry-go-round?

Do you feeling yourself thrown to one of two people side when you negotiate a curve the is ideally banked for your car’s speed? What is the direction that the pressure exerted on you by the car seat?Suppose a mass is moving in a circular route on a frictionless table as presented in figure. In the Earth’s framework of reference, there is no centrifugal pressure pulling the mass away from the center of rotation, however there is a really real force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and also Newton’s 3rd law, explain what pressure stretches the string, identifying its physical origin.

Figure 10. A fixed attached to a nail on a frictionless table moves in a one path. The force stretching the wire is real and also not fictional. What is the physical origin of the force on the string?

Problems & Exercises

(a) A 22.0 kg son is talk a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to remain on if she is 1.25 m from its center? (b) What centripetal force does she need to stay on an amusement park merry-go-round the rotates in ~ 3.00 rev/min if she is 8.00 m native its center? (c) compare each pressure with her weight.Calculate the centripetal pressure on the end of a 100 m (radius) wind turbine blade the is rotating at 0.5 rev/s. Assume the massive is 4 kg.What is the right banking angle because that a gentle turn of 1.20 km radius on a highway through a 105 km/h speed limit (about 65 mi/h), assuming everyone travels in ~ the limit?What is the ideal speed to take it a 100 m radius curve banked in ~ a 20.0° angle?(a) What is the radius the a bobsled rotate banked at 75.0° and taken in ~ 30.0 m/s, assuming the is ideally banked? (b) calculation the centripetal acceleration. (c) go this acceleration seem large to you?Part the riding a bicycle involves leaning at the exactly angle when making a turn, as checked out in number 4. To be stable, the force exerted by the ground have to be ~ above a line going with the center of gravity. The force on the bicycle wheel have the right to be resolved right into two perpendicular components—friction parallel to the road (this have to supply the centripetal force), and the vertical normal pressure (which should equal the system’s weight). (a) display that θ (as defined in the figure) is pertained to the speed v and radius the curvature r of the turn in the same means as for an ideally banked roadway—that is,  heta= an^-1fracv^2rg\; (b) Calculate θ for a 12.0 m/s turn of radius 30.0 m (as in a race).

Figure 6. 4. A bicyclist negotiating a revolve on level ground should lean in ~ the exactly angle—the capability to execute this i do not care instinctive. The force of the soil on the wheel demands to it is in on a line v the facility of gravity. The net external force on the system is the centripetal force. The upright component that the force on the wheel cancels the load of the mechanism while the horizontal component must supply the centripetal force. This process produces a relationship among the edge θ, the speed v, and also the radius the curvature r of the turn similar to that for the ideal banking the roadways.

A huge centrifuge, choose the one shown in number 5a, is used to expose aspiring astronauts to accelerations comparable to those experienced in rocket launches and atmospheric reentries. (a) at what angular velocity is the centripetal acceleration 10 g if the driver is 15.0 m native the center of rotation? (b) The rider’s cage hangs ~ above a pivot at the finish of the arm, enabling it come swing outward during rotation as displayed in figure 5b. In ~ what edge θ listed below the horizontal will certainly the cage hang once the centripetal acceleration is 10 g? (Hint: The arm gives centripetal force and also supports the weight of the cage. Draw a complimentary body chart of the forces to see what the edge θ need to be.)

Figure 5. (a) NASA centrifuge offered to topic trainees to accelerations similar to those experienced in rocket launches and also reentries. (credit: NASA) (b) driver in cage showing exactly how the cage pivots outward throughout rotation. This enables the full force exerted on the driver by the cage come be follow me its axis at every times.

Integrated Concepts. If a auto takes a banked curve at much less than the ideal speed, friction is needed to keep it native sliding towards the inside of the curve (a real problem on icy hill roads). (a) calculate the ideal speed to take a 100 m radius curve banked in ~ 15.0º. (b) What is the minimum coefficient of friction needed for a frightened driver to take the very same curve in ~ 20.0 km/h?Modern roller coasters have actually vertical loops favor the one displayed in figure 6. The radius that curvature is smaller at the peak than ~ above the sides so that the downward centripetal acceleration at the top will be greater than the acceleration as result of gravity, keeping the passenger pressed firmly into their seats. What is the speed of the roller coaster in ~ the optimal of the loop if the radius the curvature there is 15.0 m and the downward acceleration that the vehicle is 1.50 g?

Figure 6. Teardrop-shaped loops are offered in the recent roller coasters so the the radius that curvature slowly decreases come a minimum at the top. This means that the centripetal acceleration build from zero come a maximum in ~ the top and gradually to reduce again. A circular loop would reason a jolting readjust in acceleration at entry, a disadvantage uncovered long earlier in railroad curve design. With a little radius the curvature in ~ the top, the centripetal acceleration can an ext easily it is in kept higher than g so that the passengers carry out not lose contact with your seats nor execute they need seat belts to keep them in place.

Unreasonable Results. (a) calculation the minimum coefficient the friction essential for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s. (b) What is unreasonable around the result? (c) which premises room unreasonable or inconsistent?


centripetal force: any net pressure causing uniform one motion

ideal banking: the sloping that a curve in a road, whereby the angle of the slope enables the vehicle to negotiate the curve at a details speed without the aid of friction in between the tires and also the road; the net exterior force top top the automobile equals the horizontal centripetal force in the lack of friction

ideal speed: the maximum safe rate at i beg your pardon a car can turn on a curve there is no the aid of friction between the tire and also the road

ideal angle: the angle at i m sorry a automobile can revolve safely top top a steep curve, i m sorry is in proportion come the appropriate speed

banked curve: the curve in a roadway that is sloping in a manner that helps a automobile negotiate the curve

Selected remedies to troubles & Exercises

1. (a) 483 N; (b) 17.4 N; (c) 2.24 times she weight, 0.0807 times her weight

3. 4.14º

5. (a) 24.6 m; (b) 36.6 m/s2; (c) ac = 3.73 g.This does no seem as well large, yet it is clear the bobsledders feel a lot of of pressure on lock going with sharply banked turns.

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7. (a) 2.56 rad/s; (b) 5.71º

8. (a) 16.2 m/s; (b) 0.234

10. (a) 1.84; (b) A coefficient the friction this much greater than 1 is unreasonable; (c) The assumed rate is too an excellent for the tight curve.