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Consider the beam shown in (Figure 1). Suppose that w = 390 N/m . Determine the x and y components of reaction at B using scalar notation.
Express your answers using three significant figures separated by a comma.
To determine the x and y components of the reaction at B, we need to find the sum of the forces in each direction.
For the x-direction, the only force acting is the reaction at B, so the x component of the reaction at B is equal to the sum of the forces in the x-direction, which is 0 N.
For the y-direction, we need to consider the weight of the beam and the distributed load.
The weight of the beam can be calculated by multiplying the length of the beam by its weight per unit length, w. Given that w = 390 N/m, and the length of the beam is 4 m, the weight of the beam is (390 N/m) * (4 m) = 1560 N. Since the beam is symmetric, half of this weight will be supported by the reaction at B in the y-direction.
The distributed load acts downward and has a magnitude of 390 N/m. Since the length of the beam is 4 m, the total force due to the distributed load is (390 N/m) * (4 m) = 1560 N.
Therefore, the total force in the y-direction is the sum of the weight of the beam and the force due to the distributed load. Thus, in the y-direction, the reaction at B is (1560 N/2) + 1560 N = 2340 N.
Therefore, using scalar notation, the x and y components of the reaction at B are:
x-component: 0 N
y-component: 2340 N
To determine the x and y components of reaction at B, we need to analyze the forces acting on the beam. The x component of the reaction at B, denoted as F_Bx, counteracts the horizontal force exerted by the distributed load w. The y component of the reaction at B, denoted as F_By, counteracts the vertical forces exerted by the distributed load w and the weight of the beam.
To find the x component of the reaction at B (F_Bx), we need to calculate the moment about point B. Since the beam is in equilibrium, the sum of the moments about any point is zero. Therefore, taking the moment about B:
Sum of moments about B = 0
-Moment_of_w + Moment_of_F_Bx = 0
To calculate the moment of w, we can consider the distributed load as acting at the centroid of the beam. Half the length of the beam can be considered as the lever arm. The formula for the moment of a force is given by M = F * d, where F is the force and d is the lever arm distance.
Moment_of_w = w * (0.5 * L)
Now, since the beam is symmetric, the x-component of the reaction at B is equal to the moment of w:
F_Bx = Moment_of_w
To find the y component of the reaction at B (F_By), we can consider the vertical forces. The sum of the vertical forces must be equal to zero since the beam is in equilibrium:
Sum of vertical forces = 0
F_By - w * L - Weight_of_beam = 0
Since the weight of the beam is given as w_beam per unit length, we can calculate the weight of the entire beam as Weight_of_beam = w_beam * L.
Substituting this value into the equation, we get:
F_By - w * L - w_beam * L = 0
Now we can solve these two equations to find the x and y components of the reaction at B.
Please provide the value of L (length of the beam), w_beam (weight of the beam per unit length), and the distributed load w in the question for me to proceed with the calculations.
weight = 390 * 9 = 3510 Newtons
force up at ends = weight/2 = 1755 N
that is the vertical (y) component of force at B
horizontal component (x component) = (4/3)1755 = 2340 N
magnitude = (5/3)1755 = 2925 N