Three 4-kg packages A, B, and C are placed on a conveyor belt that is at rest. Between the belt and both packages A and C, the coefficients of friction are μs = 0.30 and μk = 0.20; between package B and the belt, the coefficients are μs = 0.10 and μk = 0.08. The packages are placed on the belt so that they are in contact with each other and at rest. Determine which, if any, of the packages will move and the friction force acting on each package.
Fig. P8.13
Find whether any of the package moves and the friction force acting on each package.
Answer to Problem 8.13P
The package C will
The friction force in the package C is
The packages A and B will
The friction force in the package B is
The friction force in the package A is
Explanation of Solution
Given information:
The mass of the package A, B, and C is
The static coefficient of friction between packages A and C and the belt is
The static coefficient of friction between package B and belt is
The kinetic coefficient of friction between packages A and C and belt is
The kinetic coefficient of friction between package B and belt is
Calculation:
Consider the acceleration due to gravity as
Consider Block C:
Show the free body diagram of the block C as in Figure 1.
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Find the maximum friction force
Substitute 0.30 for
The maximum friction force is greater than the friction force.
Therefore, the package C will
Therefore, the friction force in the package C is
Consider Block B:
Show the free body diagram of the block B as in Figure 2.
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Find the maximum friction force
Substitute 0.10 for
The maximum friction force is less than the friction force.
Therefore, the package B will
Find the friction force in the package B using the kinetic relation.
Substitute 0.08 for
Therefore, the friction force in the package B is
Consider Block A and B together:
Show the free body diagram of the block A and B as in Figure 3.
The normal force in package A is
The normal force in package B is
The friction force in package A is
The friction force in package B is
Find the total normal force in package A and B as follows;
Find the total friction force in package A and B as follows;
The maximum friction force in package A is
The maximum friction force in package B is
Find the maximum friction force
The maximum friction force is less than the friction force.
Therefore, the packages A and B will
Find the friction force in the package A using the kinetic relation.
Substitute 0.20 for
Therefore, the friction force in the package A is
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Chapter 8 Solutions
Vector Mechanics for Engineers: Statics
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