A 9.0 m uniform beam is hinged to a vertical wall and held horizontally by a 5.0 m cable attached to the wall 4.0 m above the hinge, as shown below. The metal of the cable has test strength of 1.00 KN (kilo-Newton), which is the maximum tension magnitude T the cable can support. That means the cable will break if T is greater than this value.
(a) (4 points) What is the angel θ ?
(b) (2 points) What is the distance along the bean between the hinge and where the cable is attached to the beam?
(c) (32 points) What is the largest beam mass m (corresponding to the test strength) the cable can support with the given configuration shown?
(d) (4 points) Find the horizontal and vertical components of the force the hinge exerts on the beam,
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To solve this problem, we will use the principles of equilibrium and trigonometry. Let's break it down step by step:
(a) To find the angle θ, we can use trigonometry. The triangle formed by the beam, cable, and wall is a right triangle. We know the lengths of two sides:
- The vertical side is 9.0 m (length of the beam).
- The horizontal side is 4.0 m (distance from the hinge to where the cable is attached to the wall).
Using the sine function, we can calculate θ:
sin(θ) = opposite/hypotenuse
sin(θ) = 4.0/9.0
Taking the inverse sine (arcsin) of both sides:
θ = arcsin(4.0/9.0)
Using a calculator, we find:
θ ≈ 25.84 degrees
Therefore, the angle θ is approximately 25.84 degrees.
(b) To find the distance along the beam between the hinge and where the cable is attached, we can use the cosine function. Again, using the right triangle formed by the beam, cable, and wall:
cos(θ) = adjacent/hypotenuse
cos(θ) = x/9.0
Solving for x:
x = 9.0 * cos(θ)
x = 9.0 * cos(25.84)
x ≈ 8.045 m
Therefore, the distance along the beam between the hinge and where the cable is attached is approximately 8.045 m.
(c) To find the largest beam mass the cable can support, we need to consider the test strength of the cable. The tension in the cable is equal to the weight of the beam, since the system is in equilibrium.
The tension in the cable is given by the equation:
Tension = mg
Where m is the mass of the beam and g is the acceleration due to gravity.
Since the cable can support a maximum tension of 1.00 kN, we can equate it to the tension in the cable, assuming the beam has the largest possible mass:
1.00 kN = m * g
Rearranging the equation to solve for m:
m = (1.00 kN) / g
The value of g is approximately 9.81 m/s².
m = (1.00 kN) / (9.81 m/s²)
m ≈ 0.102 kg
Therefore, the largest beam mass the cable can support with the given configuration is approximately 0.102 kg.
(d) The horizontal component of the force the hinge exerts on the beam is equal to the tension in the cable:
Horizontal force = Tension = 1.00 kN
The vertical component of the force the hinge exerts on the beam is equal to the weight of the beam:
Vertical force = Weight = m * g
Using the largest beam mass calculated in part (c):
Vertical force = (0.102 kg) * (9.81 m/s²)
Vertical force ≈ 1.00 N
Therefore, the horizontal component of the force the hinge exerts on the beam is 1.00 kN, and the vertical component is approximately 1.00 N.
To solve this problem, let's break it down into parts:
(a) To find the angle θ, we need to use trigonometric functions. We can use the fact that the beam is held horizontally to set up an equation.
Let's consider the vertical forces acting on the beam: the weight acting downwards and the tension in the cable acting upwards. Since the beam is held horizontally, the vertical forces must balance each other.
The weight of the beam can be calculated using the formula weight = mass × acceleration due to gravity. Assuming the beam has a uniform mass distribution, we can find the mass of the beam using the weight of the beam and the acceleration due to gravity.
Once we have the mass of the beam, we can use trigonometry to find the angle θ. We can create a right triangle with the vertical side representing the distance between the hinge and where the cable is attached to the beam (which we'll calculate in part b) and the horizontal side representing the horizontal distance of the cable attachment point from the hinge (which is given as 4.0 m).
To find the angle θ, we can use the trigonometric function tan(θ) = opposite/adjacent, where the opposite side is the vertical distance and the adjacent side is the horizontal distance.
(b) To find the distance along the beam between the hinge and where the cable is attached, we need to calculate the vertical distance.
We can use the given information that the cable is attached to the wall 4.0 m above the hinge. Since the beam is held horizontally, this vertical distance is the same as the distance between where the cable is attached to the beam and the hinge.
(c) To find the largest beam mass the cable can support, we need to consider the tension in the cable and its maximum strength.
The tension in the cable can be found by summing the horizontal forces acting on the beam. Since the beam is held horizontally, the tension in the cable must balance the horizontal component of the force exerted by the hinge on the beam.
We can find the horizontal component by using trigonometry. Since the angle θ and the vertical side (distance between the hinge and where the cable is attached) are known, we can use the trigonometric function cos(θ) = adjacent/hypotenuse to find the horizontal distance.
Once we have the horizontal component of the force exerted by the hinge, we can set it equal to the tension in the cable and solve for the maximum mass of the beam. We can use the equation weight = mass × acceleration due to gravity to convert the mass to weight, and compare it to the maximum strength of the cable.
(d) To find the horizontal and vertical components of the force the hinge exerts on the beam, we need to consider the forces acting on the beam.
The vertical component of the force exerted by the hinge can be found by summing the vertical forces acting on the beam. This includes the weight of the beam, the tension in the cable, and the vertical component of the force exerted by the hinge.
The horizontal component of the force exerted by the hinge can be found by summing the horizontal forces acting on the beam. This includes the horizontal component of the cable's tension and the horizontal component of the force exerted by the hinge.
Using the trigonometric function sin(θ) = opposite/hypotenuse, we can find the vertical component of the force exerted by the hinge. Similarly, using the trigonometric function cos(θ) = adjacent/hypotenuse, we can find the horizontal component of the force.
By considering the equilibrium of forces, i.e., that the sum of the horizontal and vertical forces must be zero, we can find the components of the force exerted by the hinge.