COLLEGE PHYSICS-CONNECT ACCESS
COLLEGE PHYSICS-CONNECT ACCESS
5th Edition
ISBN: 9781260486834
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

The figure 1 shows the free body diagram of the car.

COLLEGE PHYSICS-CONNECT ACCESS, Chapter 5, Problem 32P

Write an expression to calculate the net horizontal force.

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

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

Write an expression for the net vertical force.

  Ncosθmgfssinθ=0NcosθmgμsNsinθ=0        (II)

Here, N is the normal force, θ is the angle, fs is the static frictional force, μs is the coefficient of static friction, m the mass of the car, and g is the acceleration due to gravity.

Neglect the frictional force and re-write equation (I)

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

Neglect the frictional force and re-write equation (II)

  Ncosθmg0sinθ=0Ncosθmg=0        (IV)

Re-arrange 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.

Re-arrange equation (I) to find N

  N=mv2R(sinθ+μscosθ)        (VIII)

Re-arrange equation (II) to find N

  N=mg(cosθμssinθ)        (IX)

Compare the expression (VIII) and (IX) and rearrange the equation to find μs

  mg(cosθμssinθ)=mv2R(sinθ+μscosθ)μs=v2cosθgrsinθv2(sinθ+grcosθ)        (X)

Conclusion:

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

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

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

  μs=(20.0m/s)2cos(17.0°)(9.80m/s2)(75.0m)sin(17.0°)(20.0m/s)2(sin(17.0°)+(9.80m/s2)(75.0m)cos(17.0°))=382.5m2/s2-214.9m2/s2116.9m2/s2-702.9m2/s2=0.204

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

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Chapter 5 Solutions

COLLEGE PHYSICS-CONNECT ACCESS

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