Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 5, Problem 29P
To determine

The slowest speed the car can go around the car can go around the curve without sliding down the bank.

Expert Solution & Answer
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Answer to Problem 29P

The slowest speed the car can go around the car can go around the curve without sliding down the bank is 11.5m/s

Explanation of Solution

Write the expression for the net force acting along y direction.

Fy=Nymg+fy=0 (I)

Here, Fy is the net force acting along y direction, Ny is the normal force, m is the mass of the car, g is the acceleration due to gravity, fy is frictional force acting along y direction.

Write the expression for the net force acting along x direction.

Fx=Nxfx (II)

Here, Fx is the force acting along x direction, fx is the frictional force acting along x direction.

Write the expression for the net force acting along x direction.

Fx=mar=mv2r (III)

Here, v is the speed, and r is the radius

Equate equation II and equation III, and resolve the normal vector and force in the x and y directions,

mv2r=Nxfx=NsinθμsNcosθ (IV)

Rearrange the above equation to find v.

v=rNn(sinθμscosθ) (V)

Write the expression for the frictional force along x and y direction,

fx=μsNcosθfy=μsNsinθ (VI)

Here, μs is the coefficient of static friction, θ is the banking angle.

Substitute equation IV in equation I to find N.

Nymg+fy=Ncosθmg+μsNsinθ=0

Rearrange the above equation to find N.

N(cosθ+μssinθ)=mgN=mgcosθ+μssinθ (VII)

Substitute N from equation VII to find v.

v=rm(mgcosθ+μssinθ)(sinθμscosθ)=gr(sinθμscosθ)cosθ+μssinθ=gr(tanθμs)1+μstanθ                                                                VIII

Write the expression for the banking angle.

θ=tan1(vb2rg) (IX)

Here, θ is the banking angle, vb is the speed that a car navigate without friction.

Substitute equation IX in equation VIII to find v.

v=gr[tantan1vb2/rgμs]1+μstantan1vb2/(rg)=gr[vb2rgμs]1+μs[vb2(rg)]=vb2μsgr1+μsvb2rg (X)

Conclusion:

Substitute 15.0m/s2 for v . 0.120 for μs , 9.80m/s2 for g, 75.0m for r in equation x  to find v.

v=(15.0m/s2)0.120(9.80m/s2)(75.0m)1+0.120(15.0m/s2)9.80m/s2(75.0m)=11.5m/s

Therefore, the slowest speed the car can go around the car can go around the curve without sliding down the bank is 11.5m/s

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