Concept explainers
The keystone of an arch is the stone at the top (Fig. P14.9). It is supported by forces from its two neighbors, blocks A and B. Each block has mass m and approximate length L. What can you conclude about the force exerted by each block,
FIGURE P14.9
To show that the equilibrium conditions are satisfied by the components of the forces
Answer to Problem 9PQ
It is showed that the equilibrium conditions are satisfied by the components of the forces
Explanation of Solution
A body remains in static equilibrium if net force and torque is equal to zero.
The forces acting on the keystone are force from block A , force from block B and force of gravity, which points downward.
Write the expression for the net force acting on the keystone by block A.
Here,
Write the expression for the net force acting on the keystone by block B.
Here,
Write the expression for the force of gravity on Keystone.
Here,
It is given that Keystone is in static equilibrium.
Write the equilibrium condition for the forces along
Here,
Expand above equation using components of forces along
Rearrange above equation to get
Write the equilibrium condition for the forces along
Here,
Expand above equation using components of forces along
Assume that arch is symmetric.
Substitute
Put the above equation in equation (IX).
Substitute
Substitute
Write the expression for the torque acting on the keystone due to force
Here,
It is given that length of each block is
Write the expression for
Substitute
Use expressions
Write the expression for the torque acting on the keystone due to force
Here,
Write the expression for
Substitute
Use expressions
Thus, torque
Conclusion:
From above calculations it is clear that, net force and net torque are zero if
Each of the neighboring block supports half the weight of the keystone and the forces in the
Therefore, it is showed that the equilibrium conditions are satisfied by the components of the forces
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Chapter 14 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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