A uniform horizontal beam is attached to a vertical wall by a frictionless hinge and supported from below at an angle è = 38o by a brace that is attached to a pin. The beam has a weight of 347 N. Three additional forces keep the beam in equilibrium. The brace applies a force P to the right end of the beam that is directed upward at the angle with respect to the horizontal. The hinge applies a force to the left end of the beam that has a horizontal component H and a vertical component V. Find the magnitudes of these three forces.
For P I did 1/2 (347) = 173.5 and for v I did cos38(347). Did I start this off correctly?
You can find the solution at your previuos post
To find the magnitudes of the forces, we need to analyze the forces acting on the beam in equilibrium. Let's break down the forces and solve for each component.
1. Weight of the beam (W):
The weight of the beam is given as 347 N, acting vertically downward. This force can be represented as:
W = 347 N downward
2. Force applied by the brace (P):
You correctly used the concept of the triangle of forces. According to the problem statement, P is directed upward at an angle with respect to the horizontal. To determine the magnitude of P, you can use trigonometry. The force P can be decomposed into horizontal and vertical components as follows:
P_horizontal = P * cos(θ)
P_vertical = P * sin(θ)
3. Force applied by the hinge (F):
The hinge applies a force to the left end of the beam, which has a horizontal component (H) and a vertical component (V). Since the hinge is frictionless, there is no horizontal force applied in this case. Therefore, only the vertical component is relevant to the equilibrium of forces.
Based on the problem statement, the vertical component of the hinge force should balance the weight of the beam since the beam is in equilibrium.
Here are the calculations:
P_horizontal = P * cos(θ) (correct)
P_vertical = P * sin(θ)
To find P and V, we can set up equations using the equilibrium condition:
Sum of vertical forces = 0:
P_vertical + V - W = 0
Substituting the given values:
P * sin(θ) + V - 347 N = 0
To find V, we need to use the value of θ given in the problem statement.
To calculate P, you made an error. Instead of halving the weight, let's solve the equation for P:
P = (347 N - V) / sin(θ)
Substituting the known values:
P = (347 N - V) / sin(38°)
Now, we have two equations:
P = (347 N - V) / sin(38°)
P_vertical + V = 347 N
From here, you can solve these equations simultaneously to find the values of P and V.