A uniform beam AB of mass 65.3kg rests against the smooth vertical wall at A, and the rough horizontal floor at B, at an angle 45deg. The coefficient of static friction between the beam and the horizontal floor is ì=0.242.
Calculate the magnitude of the horizontal force P, in N, such that the end B has impending motion to the left.
To calculate the magnitude of the horizontal force P required for impending motion of end B to the left, we need to consider the forces acting on the beam.
First, let's identify the forces acting on the beam AB:
1. Weight of the beam (W): This force acts vertically downward and is equal to the mass of the beam multiplied by the acceleration due to gravity (9.8 m/s^2).
2. Normal force (N): This force acts perpendicular to the contact surface between the beam and the floor and is equal to the weight of the beam (W).
3. Friction force (F): This force acts parallel to the contact surface between the beam and the floor and opposes the impending motion of end B.
Now, let's break down the forces and calculate the magnitude of the horizontal force P:
1. Weight of the beam (W): W = mass × gravitational acceleration
W = 65.3 kg × 9.8 m/s^2
W ≈ 639.94 N
2. Normal force (N): N = weight of the beam (W)
N ≈ 639.94 N
3. Friction force (F): F = coefficient of static friction × normal force
F = μ × N
F = 0.242 × 639.94 N
F ≈ 154.91 N
Since the beam is in equilibrium, the horizontal force P must be equal in magnitude but opposite in direction to the friction force (F). Therefore:
Magnitude of horizontal force P = |F| ≈ 154.91 N
Hence, the magnitude of the horizontal force P required for impending motion of end B to the left is approximately 154.91 N.