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.
I do not know how to start this.
Net torque about the hinge is
mg•(L/2)- P(y) •L = 0,
( Forces H and V have zero moment about this point)
P(y) = mg/2 =347/2 =173.5 N.
P(y)/P(x) =tanα.
P(x) = P(y)/tanα=173.5/tan38 =222 N.
P(y)/P =sin α,
P =P(y)/sin α =173.5/sin38 =282 N.
As the system is in equilibrium, horizontal forces are balanced:
Force H acts into the wall (to the left)= P(x) (to the right).
H = P(x) = 222 N.
Vertical forces are also in balance
mg – V - P(y) =0
V = mg – P(y) =347 – 173.5 =173.5 N.
To start solving this problem, we need to analyze the forces acting on the horizontal beam in equilibrium. We can break down the forces into their horizontal and vertical components.
Let's consider the horizontal forces first. In equilibrium, the sum of the horizontal forces must be zero.
1. The brace applies a force P to the right end of the beam, directed upward at an angle with respect to the horizontal. To find the horizontal component of this force, we can use the trigonometric relationship:
Horizontal component of P = P * cos(angle)
2. The hinge applies a horizontal force H. Since there is no other horizontal force mentioned in the problem, the horizontal component of H balances the horizontal components of the other forces.
Now, let's consider the vertical forces. In equilibrium, the sum of the vertical forces must also be zero.
1. The weight of the beam applies a force W = 347 N downwards.
2. The brace applies a vertical force P upward at an angle with respect to the horizontal. To find the vertical component of this force, we can use the trigonometric relationship:
Vertical component of P = P * sin(angle)
3. The hinge applies a vertical force V. Since there is no other vertical force mentioned in the problem, the vertical component of V balances the vertical components of the other forces.
To solve for the magnitudes of these three forces (P, H, and V), we need to use the given information. In the problem statement, it says the brace is attached at an angle of 38 degrees (è = 38o) and the weight of the beam is 347 N.
Now, we can set up equations for the horizontal and vertical components of the forces:
Horizontal forces: H + P * cos(angle) = 0
Vertical forces: -W + P * sin(angle) + V = 0
By substituting the given values into these equations, we can solve for the unknown quantities P, H, and V.