Concept explainers
The coefficients of friction between the 30-lb block and the 5-lb platform BD are μs = 0.50 and μk = 0.40. Determine the accelerations of the block and of the platform immediately after wire AB has been cut.
Fig. P16.20
Calculate the accelerations of the block and of the platform immediately after wire AB has been cut
Answer to Problem 16.20P
The accelerations of the block and of the platform immediately after wire AB has been cut
Explanation of Solution
Given information:
The weight of the block
The weight of the platform BD
The coefficients of static friction
The coefficients of kinetic friction
The length of the platform BD (L) is
Calculation:
Consider the acceleration due to gravity (g) is
Consider the block does not slide relative to the platform.
Show the free body and kinetic diagram of the platform and block as in Figure 1.
Here,
Refer to Figure 1.
Calculate the forces at the tangent:
Substitute
Calculate the acceleration of the block and of the platform (a):
Substitute and
Check whether or not the block will slide relative to the platform.
Show the free body and kinetic diagram of the block as in Figure 2.
Here, F is the horizontal force of the block and N is the vertical force of the block.
Refer to Figure 2
Calculate the horizontal forces by applying the equation of equilibrium:
Substitute
Substitute
Calculate the vertical forces by applying the equation of equilibrium:
Substitute
Substitute
Calculate the force of the block
Substitute 0.50 for
Since, the block will slide.
Consider the block slides relative to the platform.
Calculate the Equations of motion for block:
Show the free body and kinetic diagram of the block as in Figure 3.
Here,
Refer to Figure 3
Calculate the vertical forces by applying the equation of equilibrium:
Calculate the horizontal forces by applying the equation of equilibrium:
Substitute
Substitute 0.40 for
Calculate the Equations of motion for platform:
Show the free body and kinetic diagram of the platform as in Figure 4.
Here,
Refer to Figure 4
Calculate the forces at the tangent:
Substitute
If contact is maintained between block and platform.
Therefore,
Calculate the vertical force of the block (N):
Substitute
Calculate the acceleration of the block at x direction
Substitute
Calculate the acceleration of the block at y direction
Substitute
Calculate the accelerations of the block
Substitute
Calculate the angle of the block
Substitute
Calculate the accelerations of the platform
Substitute
Calculate the angle of the platform
Substitute
Hence, the accelerations of the block and of the platform immediately after wire AB has been cut
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