Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 12, Problem 77PQ
To determine

The coefficient of static friction between the car’s tires and the road.

Expert Solution & Answer
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Answer to Problem 77PQ

The coefficient of static friction between the car’s tires and the road is 0.98 .

Explanation of Solution

The force of static friction on the automobile must act forward and then more and more inward on the tires, to produce both tangential acceleration and to maintain the centripetal acceleration.

Write the equation for the tangential force.

  Ft=ma                                                                                                                  (I)

Here, Ft is the magnitude of the tangential force, m is the mass of the automobile and a is the magnitude of the tangential acceleration.

The centripetal force is provided by the radially inward component of force.

Write the equation for the centripetal force.

  Fr=mac                                                                                                                (II)

Here, Fr is the centripetal force and ac is the centripetal acceleration of the car’s tires.

Write the equation for ac .

  ac=v2r                                                                                                                 (III)

Here, v is the translational speed and r is the radius of the semicircular curve.

Write the equation for the angular speed.

  ω=vr

Here, ω is the angular speed.

Multiply both numerator and denominator of right hand side of equation (III) and put the above equation in it.

  ac=v2r2r=ω2r

Put the above equation in equation (II).

  Fr=mω2r                                                                                                            (IV)

The centripetal force increases with time as the speed increases and it reaches its maximum value when the automobile has travelled 3/4 into a semicircular curve or in other words, when its angle has increased by 3π/4 .

Write the equation for ω in terms of initial angular velocity.

  ω2=ω02+2αΔθ

Here, ω0 is the initial angular velocity, α is the angular acceleration and Δθ is the angular displacement.

Put the above equation in equation (IV).

  Fr=m(ω02+2αΔθ)r

The automobile starts from rest so that its initial angular velocity must be zero.

Substitute 0 for ω0 and 3π/4 for Δθ in the above equation.

  Fr=m(0+2α(3π4))r=m(3π2)rα                                                                                         (V)

Write the equation for the linear acceleration in terms of angular acceleration.

  a=rα

Put the above equation in equation (V).

  Fr=m(3π2)a                                                                                                      (VI)

The force of friction when the car slips is equal to the magnitude of the total force found using the Pythagorean theorem.

fs=Ft2+Fr2 (VII)

Here, fs is the force of friction.

Write the equation for fs .

  fs=μsFN                                                                                                          (VIII)

Here, μs is the coefficient of static friction and FN is the normal force

The normal force is equal to the weight of the car.

Write the equation for FN .

  FN=mg

Here, g is the acceleration due to gravity.

Put the above equation in equation (VIII).

  fs=μsmg                                                                                                            (IX)

Put equations (I), (VI) and (IX) in equation (VII) and rewrite the equation for μs .

  μsmg=(ma)2+(m(3π2)a)2=m2a2+m2a2(3π2)2=ma1+(3π2)2μs=ag1+(3π2)2                                                                            (X)

Conclusion:

Given that the tangential acceleration is 2.00 m/s2 . The value of acceleration due to gravity is 9.81 m/s2 .

Substitute 2.00 m/s2 for a and 9.81 m/s2 for g in equation (X) to find μs .

  μs=2.00 m/s29.81 m/s21+(3π2)2=0.98

Therefore, the coefficient of static friction between the car’s tires and the road is 0.98 .

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Chapter 12 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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