The mass of a beam can be found by calculating the product of the linear density
ID: 1908554 • Letter: T
Question
The mass of a beam can be found by calculating the product of the linear density and the length of the beam or the mass density times the volume of the beam. In the diagram a square beam 9 meters long ( L ) with a mass density of 640 kg/m 3 is supported by a spring scale and is hinged at the left. The spring scale is attached to the beam at a distance of 2 meters (d) from the left side of the beam. The height of the square beam ( h ) is 60 cm. A weight of 7000 newtons is suspended at the far right end of the beam.
1- Find the mass of the beam.
2-Find the linear density of the beam
3-A moment is valued mathematically as the product of the force and the moment arm. The moment arm is the perpendicular distance from the point of rotation, to the line of action of the force. The moment may be thought of as a measure of the tendency of the force to cause rotation about an imaginary axis through a point. In this case the moment arm is L as the beam is horizontal.and the weight is acting vertically.
4-Find the total clockwise moment about A.
5-Find the tension in the spring scale
6-A force in mechanics is a concept that has both a size and a direction. The net force acting on an object is the sum of the forces onto the object, taking into account both their sizes, and their directions. For example, if an object has two forces acting on it, with equal sizes but in opposite directions, then the net force will be zero (technically: a null vector). For a beam in static equilibrium the sum of the forces in the X, Y and Z axis must be zero Find the horizontal component of the force at the hinge.
7-Find the vertical component of the force at the hinge.
Explanation / Answer
volume = area*length = h^2*L = 0.6^2 *9 = 3.24 m3
1. Mass = density*volume = 640*3.24 = 2073.6 kg
2. linear density = Mass/L = 2073.6/9 = 230.4 kg/m
3. Moment due to hanging weight = Hanging Weight*L = 7000*9 = 63000 Nm
4.
weight of beam W = mass*g = 2073.6*9.81 = 20342 N
This weight acts from the centre of the beam.
Moment due to weight = 20342*(L/2) = 20342*9/2 = 91539 Nm
Total clockwise moment about A is = 63000 + 91539 = 154539 Nm
5.
Balancing moments about A, 154539 = T*d
154539 = T*2
T = 77269.5 N
6.
Since there are no forces acting on the beam in horizontal direction, horizontal component of force at hinge = 0
7. Balancing forces in vertical direction, Fa + T = Weight of beam + hanging weight
Fa + 77269.5 = 20342 + 7000
Fa = -49927.5 N
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