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F c = m v 2 r F c = mr ω 2 } , size 12{ left none matrix { F rSub { size 8{c} } =m { {v rSup { size 8{2} } } over {r} } {} ##F rSub { size 8{c} } = ital "mr"ω rSup { size 8{2} } } right rbrace ,} {}
m v 2 r = μ s mg . size 12{m { {v rSup { size 8{2} } } over {r} } =μ rSub { size 8{s} } ital "mg"} {}

We solve this for μ s size 12{μ rSub { size 8{s} } } {} , noting that mass cancels, and obtain

μ s = v 2 rg . size 12{μ rSub { size 8{s} } = { {v rSup { size 8{2} } } over { ital "rg"} } } {}

Solution for (b)

Substituting the knowns,

μ s = ( 25.0 m/s ) 2 ( 500 m ) ( 9 . 80 m/s 2 ) = 0 . 13 . size 12{μ rSub { size 8{s} } = { { \( "25" "." 0" m/s" \) rSup { size 8{2} } } over { \( "500"" m" \) \( 9 "." "80 m/s" rSup { size 8{2} } \) } } =0 "." "13"} {}

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

Discussion

We could also solve part (a) using the first expression in F c = m v 2 r F c = mr ω 2 } , size 12{ left none matrix { F rSub { size 8{c} } =m { {v rSup { size 8{2} } } over {r} } {} ##F rSub { size 8{c} } = ital "mr"ω rSup { size 8{2} } } right rbrace ,} {} because m , size 12{m,} {} v , size 12{v,} {} and r size 12{r} {} are given. The coefficient of friction found in part (b) is much smaller than is typically found between tires and roads. The car will still negotiate the curve if the coefficient is greater than 0.13, because static friction is a responsive force, being able to assume a value less than but no more than μ s N size 12{μ rSub { size 8{g} } N} {} . A higher coefficient would also allow the car to negotiate the curve at a higher speed, but if the coefficient of friction is less, the safe speed would be less than 25 m/s. Note that mass cancels, implying that in this example, it does not matter how heavily loaded the car is to negotiate the turn. Mass cancels because friction is assumed proportional to the normal force, which in turn is proportional to mass. If the surface of the road were banked, the normal force would be less as will be discussed below.

In the given figure, a car is shown from the back, which is turning to the left. The weight, w, of the car is shown with a down arrow and N with an up arrow at the back of the car. At the right rear wheel, centripetal force is shown along with its equation formula in a leftward horizontal arrow. The free-body diagram shows three vectors, one upward, depicting N, one downward, depicting w, and one leftward, depicting centripetal force.
This car on level ground is moving away and turning to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. A minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway.

Let us now consider banked curves , where the slope of the road helps you negotiate the curve. See [link] . The greater the angle θ size 12{θ} {} , the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle θ size 12{θ} {} is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for θ size 12{θ} {} for an ideally banked curve and consider an example related to it.

For ideal banking    , the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions.

[link] shows a free body diagram for a car on a frictionless banked curve. If the angle θ size 12{θ} {} is ideal for the speed and radius, then the net external force will equal the necessary centripetal force. The only two external forces acting on the car are its weight w size 12{w} {} and the normal force of the road N size 12{N} {} . (A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force.) These two forces must add to give a net external force that is horizontal toward the center of curvature and has magnitude mv 2 /r size 12{"mv" rSup { size 8{2} } "/r"} {} . Because this is the crucial force and it is horizontal, we use a coordinate system with vertical and horizontal axes. Only the normal force has a horizontal component, and so this must equal the centripetal force—that is,

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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