Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 5, Problem 32P
To determine

The minimum coefficient of static friction needed if the car is to navigate the curve without slipping.

Expert Solution & Answer
Check Mark

Answer to Problem 32P

The minimum coefficient of static friction needed if the car is to navigate the curve without slipping is 0.204.

Explanation of Solution

Figure 1 shows the free body diagram of the car.

Physics, Chapter 5, Problem 32P

Write an expression to calculate the net horizontal force.

Nsinθ+fscosθ=maxNsinθ+μsNcosθ=m(v2r) (I)

Here, the normal force is N, the angle is θ, the static frictional force is fs, the coefficient of static friction is μs, the mass of the car is m, the horizontal acceleration is ax, the speed is v and the radius of path is r.

Write an expression for the net vertical force.

Ncosθmgfssinθ=0NcosθmgμsNsinθ=0 (II)

Here, the acceleration due to gravity is g.

Neglect the frictional force and rewrite equation (I).

Nsinθ+(0)cosθ=m(v2r)Nsinθ=m(v2r) (III)

Neglect the frictional force and rewrite equation (II).

Ncosθmg(0)sinθ=0Ncosθmg=0 (IV)

Rearrange equation (IV) to find N.

N=mgcosθ (V)

Substitute equation (V) in equation (III). 

(mgcosθ)sinθ=m(v2r)gtanθ=(v2r) (VI)

Rewrite equation (VI) to find θ.

θ=tan1(v2gr) (VII)

Now consider the presence of frictional force.

Rearrange equation (I) to find N.

N=mv2R(sinθ+μscosθ) (VIII)

Rearrange equation (II) to find N.

N=mgcosθμssinθ (IX)

Compare equation (VIII) and (IX).

mgcosθμssinθ=mv2r(sinθ+μscosθ)gcosθμssinθ=v2r(sinθ+μscosθ) . (X)

Rearrange equation (X) to find μs.

μs=v2cosθgrsinθv2sinθ+grcosθ . (XI)

Conclusion:

Substitute 15.0m/s for v, 9.80m/s2 for g, and 75.0m for r in equation (VII) to find θ.

θ=tan1((15.0m/s)2(9.80m/s2)(75.0m))=tan1(0.306)=17.0°

Substitute 20.0m/s for v, 9.80m/s2 for g, 75.0m for r, and 17.0° for θ in equation (XI) to find μs.

μs=(20.0m/s)2cos(17.0°)(9.80m/s2)(75.0m)sin(17.0°)(20.0m/s)2sin(17.0°)+(9.80m/s2)(75.0m)cos(17.0°)=382.5m2/s2214.9m2/s2116.9m2/s2+702.9m2/s2=0.204

Thus, the minimum coefficient of static friction needed if the car is to navigate the curve without slipping is 0.204.

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