In Fig. 12-46, suppose the length L of the uniform bar is 2.95 m and its weight is 164 N. Also, let the block's weight W = 468 N and the angle θ = 33.8°. The wire can withstand a maximum tension of 520 N. (a) What is the maximum possible distance x before the wire breaks? With the block placed at this maximum x, what are the (b) horizontal and (c) vertical components of the force on the bar from the hinge at A?its like a right angle triangle, with beam at the base, wall is along y axis ,block is between wall and centre of mass of beam and cable is the hypotenuse
Set the moment about the hinge equal to zero, with a cable tension equal to the maximum value, 520 N. The torque due to the weight will be 468x Newton-meters.
Solve for x. Once you know x, apply a force balance on the entire beam to get the vertical and horizontal forces at the hinge
To determine the maximum possible distance x before the wire breaks, we need to consider the forces acting on the system.
(a) First, let's analyze the forces acting on the block. We have the weight W acting vertically downwards and the tension in the wire T acting upwards at an angle θ with the vertical axis. The vertical component of T can be given by W = T * sin(θ). Given that W = 468 N and θ = 33.8°, we can rearrange this equation to find T:
T = W / sin(θ)
Substituting the given values, T = 468 N / sin(33.8°) = 852 N.
Now, we need to analyze the forces acting on the bar. We have the weight of the bar (164 N) acting downwards at its center of mass. The vertical component of the force from the hinge at A (Fy) can balance this weight:
Fy = 164 N
Next, we consider the horizontal component of the tension in the wire (Tx). To find Tx, we can use the equation Tx = T * cos(θ). Given that θ = 33.8° and T = 852 N, we can substitute these values to find Tx:
Tx = 852 N * cos(33.8°) = 710 N.
(b) With the block placed at the maximum distance x, the horizontal component of the force on the bar from the hinge at A is equal to Tx. Therefore, the horizontal force on the bar is 710 N.
(c) The vertical component of the force on the bar from the hinge at A remains unchanged and equal to Fy, which is 164 N. Therefore, the vertical force on the bar is 164 N.
To summarize:
(a) The maximum possible distance x before the wire breaks is dependent on the tension in the wire. We found this tension to be 852 N.
(b) With the block placed at this maximum x, the horizontal component of the force on the bar from the hinge at A is 710 N.
(c) The vertical component of the force on the bar from the hinge at A is 164 N.