Principles of Physics: A Calculus-Based Text
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 5, Problem 54P
To determine

The reason for the given situation in the problem is not possible.

Expert Solution & Answer
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Answer to Problem 54P

The speed of the child is too large, so static friction does not have the strength to keep the child in place on the incline. Hence the situation is impossible.

Explanation of Solution

Assume that, the friction points up the incline, the net force is directed left towards the centre of the circular path in which the child travels, and a centripetal force needed for the circular motion is directed towards the centre of the circular path. The radius of the path of child is R=dcosθ.

Free body diagram of the child is shown in Figure.

Principles of Physics: A Calculus-Based Text, Chapter 5, Problem 54P

Write the expression for net force in the horizontal direction, from the free body diagram.

    fscosθnsinθ=mv2R        (I)

Here, fs is the force due to static friction, n is the normal force, θ is the angle of inclination, m is the mass, v is the speed, and R is the radius of the circular path.

Write the expression for net force in the horizontal direction, from the free body diagram.

    fssinθ+ncosθ=mg        (II)

Here, g is the acceleration due to gravity.

Multiply equation (I) by cosθ.

    fscos2θnsinθcosθ=mv2Rcosθ        (III)

Multiply equation (II) by sinθ.

    fssin2θ+nsinθcosθ=mgsinθ        (IV)

Add equation (III) and (IV), and solve for fs.

    fscos2θnsinθcosθ+fssin2θ+nsinθcosθ=mgsinθ+mv2Rcosθfscos2θ+fssin2θ=mgsinθ+mv2Rcosθfs(cos2θ+sin2θ)=mgsinθ+mv2Rcosθfs=mgsinθ+mv2Rcosθ        (V)

Multiply equation (I) by sinθ.

    fssinθcosθ+nsin2θ=mv2Rsinθ        (VI)

Multiply equation (II) by cosθ.

    fssinθcosθ+ncos2θ=mgcosθ        (VII)

Add equation (VI) and (VII), and solve for n.

    fssinθcosθ+nsin2θ+fssinθcosθ+ncos2θ=mgcosθmv2Rsinθnsin2θ+ncos2θ=mgcosθmv2Rsinθn(sin2θ+cos2θ)=mgcosθmv2Rsinθn=mgcosθmv2Rsinθ        (VIII)

Equation (V) and (VIII) are consistent only when fsμsn.

Use equation (V) and (VIII) in fsμsn, and solve for v.

    mgsinθ+mv2Rcosθμs(mgcosθmv2Rsinθ)v2R(cosθ+μssinθ)g(μscosθsinθ)v2(cosθ+μssinθ)gR(μscosθsinθ)        (IX)

Here, μs is the coefficient of static friction.

Use dcosθ for R in the equation (IX).

    v2(cosθ+μssinθ)gdcosθ(μscosθsinθ)v2gdcosθ(μscosθsinθ)(cosθ+μssinθ)vgdcosθ(μscosθsinθ)(cosθ+μssinθ)        (X)

Here, d is the distance to the child from the centre of the cone.

If v is too large, this situation is cannot exist.

Conclusion:

Substitute, 9.80m/s2 for g, 5.32m for d, 20.0° for θ, and 0.700 for μs in the equation (X).

    v(9.80m/s2)(5.32m)cos20.0°[(0.700)cos20.0°sin20.0°](cos20.0°+μssin20.0°)3.62m/s

Since, the speed of the child is too large, so static friction does not have the strength to keep the child in place of the incline. Hence the situation is impossible.

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Principles of Physics: A Calculus-Based Text

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