Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 2.3, Problem 2.50P

Two cables are tied together at C and are loaded as shown. Determine the range of values of P for which both cables remain taut.

Chapter 2.3, Problem 2.50P, Two cables are tied together at C and are loaded as shown. Determine the range of values of P for

Fig. P2.49 and P2.50

Expert Solution & Answer
Check Mark
To determine

The range of values of the force P for which the cables CA and CB remain taut.

Answer to Problem 2.50P

The range of values of the force P for which the cables CA and CB remain taut is 179.3N<P<669N_.

Explanation of Solution

In the arrangement in Fig. P2.50, the forces acting at the point c are one of magnitude 200N and the other of unknown magnitude P.

The free body diagram corresponding to the given arrangement is shown in Figure 1.

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 2.3, Problem 2.50P

When the system is in equilibrium, the total forces acting on the system will be zero. Moreover, the equilibrium condition imply that the components of the forces along x direction and y direction will be separately equal to zero.

Write the expression for the equilibrium state along x direction and y direction.

ΣFx=0 (I)

ΣFy=0 (II)

Here, Fx and Fy are the components of the forces along x and y direction respectively.

Apply the condition in equation (I) to the given system.

TCAsin30°+TCBsin30°+Pcos45°200N=0 (III)

Apply the condition in equation (II) to the given system.

TCAcos30°+TCBcos30°Psin45°=0 (IV)

Write the condition for the cable CA to remain taut.

TCA>0 (V)

Write the condition for the cable CA to remain taut.

TCB>0 (VI)

The condition for both the cables to remain taut is that, there should be a tension greater than zero in each of the cables.

Conclusion:

Set TCA=0 in equation (III) and (IV).

(0)sin30°+TCBsin30°+Pcos45°200N=0TCBsin30°+Pcos45°200N=0 (VII)

(0)cos30°+TCBcos30°Psin45°=0TCBcos30°Psin45°=0 (VIII)

Add equation (VII) and (VIII).

TCB(sin30°+cos30°)+P(sin45°cos45°)200N=0TCB(sin30°+cos30°)200N=0 (IX)

Solve equation (IX) for TCB.

TCB=200N(sin30°+cos30°)=146.41N

Solve equation (VIII) for P and substitute 146.41N for TCB to find the magnitude of P.

P=TCBcos30°sin45°=(146.41N)cos30°sin45°=179.3N (X)

Set TCB=0 in equation (III) and (IV).

TCAsin30°+(0)sin30°+Pcos45°200N=0TCAsin30°+Pcos45°200N=0 (XI)

TCAcos30°+(0)cos30°Psin45°=0TCAcos30°Psin45°=0 (XII)

Add equation (XI) and (XII).

TCA(cos30°sin30°)+P(cos45°sin45°)200N=0TCA(cos30°sin30°)200N=0 (XIII)

Solve equation (XIII) for TCA.

TCA=200N(cos30°sin30°)=546.41N

Solve equation (XII) for P and substitute 546.41N for TCA to find the magnitude of P.

P=TCAcos30°sin45°=(546.41N)cos30°sin45°=669.2N (XIV)

The magnitude of P obtained in equation (X) and (XIV) are the range of values of P for which both the cables remain taut.

Therefore, the range of values of the force P for which the cables CA and CB remain taut is 179.3N<P<669.2N_.

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Chapter 2 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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