Concept explainers
A uniform beam of mass m is inclined at an angle θ to the horizontal. Its upper end (point P) produces a 90° bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (Fig. P12.35). Let μs represent the coefficient of static friction between beam and floor. Assume μs is less than the cotangent of θ. (a) Find an expression for the maximum mass M that can be suspended from the top before the beam slips. Determine (b) the magnitude of the reaction force at the floor and (c) the magnitude of the force exerted by the beam on the rope at P in terms of m, M, and μs.
Figure P12.35
(a)
The expression for the maximum mass that can be suspended from the top surface before the beam starts slip.
Answer to Problem 35AP
The expression for the maximum mass that can be suspended from the top surface before the beam starts slip is
Explanation of Solution
The mass of the beam is
The following figure shows the force diagram of the beam.
Figure-(I)
Formula to calculate the frictional force acting on the base of the beam is,
Here,
Formula to calculate the net torque about the point
Here,
Rearrange the above equation to find
Formula to calculate the net vertical forces is,
Rearrange the above equation to find
Formula to calculate the net horizontal force is,
Substitute
Further simplify the above equation to find
Conclusion:
Therefore, the expression for the maximum mass that can be suspended from the top surface before the beam starts slip is
(b)
The magnitude of the reaction force at the floor.
Answer to Problem 35AP
The magnitude of the reaction force at the floor is
Explanation of Solution
The mass of the beam is
Formula to calculate the net reaction force acting in the floor is,
Here,
Substitute
Conclusion:
Therefore, the magnitude of the reaction force at the floor is
(c)
The magnitude of the force exerted by the beam in the rope at
Answer to Problem 35AP
The magnitude of the force exerted by the beam in the rope at
Explanation of Solution
The mass of the beam is
Since the coefficient of static friction of the floor is less than cotangent of the angle
Substitute
Formula to calculate the magnitude of the net force exerted by the beam in the rope at point
Here,
Substitute
Substitute
Conclusion:
Therefore, the magnitude of the force exerted by the beam in the rope at
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