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 4, Problem 50P

(a)

To determine

The expression for the tension in each section of string.

(a)

Expert Solution
Check Mark

Answer to Problem 50P

The expression for the tension in each section of string is T1=2mgsinθ1,T2=mgsin[tan1(12tanθ1)],T3=2mgtanθ1.

Explanation of Solution

The free body diagram for the system is given by figure 1.

Principles of Physics: A Calculus-Based Text, Chapter 4, Problem 50P

Write the expression for the net force from Newton’s second law.

T2cosθ2T1cosθ1=0        (I)

T1sinθ1T2sinθ2mg=0        (II)

T2cosθ2T3=0        (III)

T2sinθ2mg=0        (IV)

Here, T1,T2,T3 are the tensions, θ1,θ2 are the angles, m is the mass and g is the gravitational acceleration.

Conclusion:

Substitute, mg for T2sinθ2 in Equation (II) to find T1.

  T1sinθ1mgmg=0T1=2mgsinθ1

Substitute, T3 for T2cosθ2, (2mgsinθ1) for T1 in Equation (I) to find T3.

T3(2mgsinθ1)cosθ1=0T3=2mgtanθ1

Divide equation (IV) by (III) and substitute (2mgtanθ1) for T3 to find θ2.

T2sinθ2T2cosθ2=mgT3T2sinθ2T2cosθ2=mg(2mgtanθ1)θ2=tan1(tanθ12)

Substitute, tan1(tanθ12) for θ2 in Equation (IV) to find T2.

T2sin[tan1(tanθ12)]mg=0T2=mgsin[tan1(tanθ12)]

Thus, the expression for the tension in each section of string is T1=2mgsinθ1,T2=mgsin[tan1(12tanθ1)],T3=2mgtanθ1.

(b)

To determine

The expression for the angle θ2 in terms of θ1.

(b)

Expert Solution
Check Mark

Answer to Problem 50P

The expression for the angle θ2 in terms of θ1 is θ2=tan1(tanθ12)_.

Explanation of Solution

Write the expression for the net force from Newton’s second law.

T2cosθ2T1cosθ1=0        (I)

T1sinθ1T2sinθ2mg=0        (II)

T2cosθ2T3=0        (III)

T2sinθ2mg=0        (IV)

Conclusion:

Divide equation (IV) by (III) and substitute (2mgtanθ1) for T3 to find θ2.

T2sinθ2T2cosθ2=mgT3T2sinθ2T2cosθ2=mg(2mgtanθ1)θ2=tan1(tanθ12)

Thus, the expression for the angle θ2 in terms of θ1 is θ2=tan1(tanθ12)_.

(c)

To determine

The expression for the distance between the endpoints of the string.

(c)

Expert Solution
Check Mark

Answer to Problem 50P

The expression for the distance between the endpoints of the string is D=L5[(2cosθ1)+[2cos{tan1(12tanθ1)}]+1]_.

Explanation of Solution

Write the expression for the total length.

L=5l        (V)

Here, L is the total length and l is the length of each portion.

Write the expression for the distance.

D=2lcosθ1+2lcosθ2+l        (VI)

Conclusion:

Substitute, (L/5) for l, tan1(tanθ12) for θ2 in Equation (VI) to find D.

  D=2(L/5)cosθ1+2(L/5)cos[tan1(tanθ12)]+(L/5)=L5[(2cosθ1)+[2cos{tan1(12tanθ1)}]+1]

Thus, the expression for the distance between the endpoints of the string is D=L5[(2cosθ1)+[2cos{tan1(12tanθ1)}]+1]_.

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Chapter 4 Solutions

Principles of Physics: A Calculus-Based Text

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