Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 5, Problem 31P

(a)

To determine

The frictional force on the car.

(a)

Expert Solution
Check Mark

Answer to Problem 31P

The frictional force on the car is 2300N_.

Explanation of Solution

Given that the radius of the curved path is 410m, the speed of the car is 32m/s, the banking angle of the road is 5.0°, and the mass of the car is 1400kg.

Let the the x-axis point horizontally toward the the center of curvature and y-axis point upward.

The Free-body diagram of the car in the given situation is shown in Figure 1.

Physics, Chapter 5, Problem 31P

Write the condition for equilibrium of forces in the x and y directions.

Fx=0 (I)

Fy=0 (II)

Apply the condition in equation (II) to the free-body diagram of the car.

Ncosθmgfsinθ=0 (III)

Here, N is the magnitude of normal force, θ is the angle of banking, m is the mass of the car, g is the acceleration due to gravity, and f is the magnitude of the frictional force.

Solve equation (III) for N.

N=fsinθ+mgcosθ (IV)

Apply the condition in equation (I) to the free-body diagram of the car.

Nsinθ+fcosθmv2r=0 (V)

Here, mv2/r represents the centripetal force.

Use equation (IV) in (V) and solve for f.

(fsinθ+mgcosθ)sinθ+fcosθmv2r=0(fsinθ+mg)(sinθcosθ)+fcosθ=mv2rf=(mv2/r)mg(sinθcosθ)(sinθ(sinθcosθ)+cosθ)=m(v2cosθrgsinθ) (VI)

Conclusion:

Substitute 32m/s for v, 5.0° for θ, 410m for r, 9.80m/s2 for g, and 1400kg for m in equation (VI) to find f.

f=(1400kg)((32m/s)2cos5.0°410m(9.80m/s2)sin5.0°)=2287N2300N

Therefore, the frictional force on the car is 2300N_.

(b)

To determine

The speed with which one must drive the car so that the force of friction is zero.

(b)

Expert Solution
Check Mark

Answer to Problem 31P

The speed with which one must drive the car so that the force of friction is zero is 19m/s_.

Explanation of Solution

Write the expression for the force of friction for the given track.

f=m(v2cosθrgsinθ)

The force of friction to be zero, the term f in the above equation has to be replaced by 0.

0=m(v2cosθrgsinθ) (VII)

Solve equation (VII) for v.

v2=rgsinθcosθv=rgtanθ (VIII)

Conclusion:

Substitute 410m for r, 9.80m/s2 for g, and 5.0° for θ in equation (VIII) to find v.

v=(410m)(9.80m/s2)(tan5.0°)=18.74m/s19m/s

Therefore, the speed with which one must drive the car so that the force of friction is zero is 19m/s_.

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