A uniform steel beam of length 5.50 m has a weight of 4.50 103 N. One end of the beam is bolted to a vertical wall. The beam is held in a horizontal position by a cable attached between the other end of the beam and a point on the wall. The cable makes an angle of 25.0° above the horizontal. A load whose weight is 12.0 103 N is hung from the beam at a point that is 4.60 m from the wall.
(a) Find the magnitude of the tension in the supporting cable.
N
(b) Find the magnitude of the force exerted on the end of the beam by the bolt that attaches the beam to the wall.
N
To find the magnitude of the tension in the supporting cable and the force exerted on the end of the beam by the bolt, we can use the principles of equilibrium.
First, let's analyze the forces acting on the beam:
1. Weight of the beam: The weight of the beam acts at its center of mass (midpoint) and has a magnitude of 4.50 * 10^3 N. We can consider this force acting vertically downward.
2. Tension in the supporting cable: This force is directed along the cable and is responsible for keeping the beam in a horizontal position. Let's denote the tension as T.
3. Force exerted by the bolt: This force is exerted perpendicular to the wall at the point where the beam is attached. Let's denote this force as F.
4. Weight of the load: The load hanging from the beam has a weight of 12.0 * 10^3 N.
To solve for the unknowns:
(a) Magnitude of the tension in the supporting cable (T).
(b) Magnitude of the force exerted on the end of the beam by the bolt (F).
Let's start with finding the magnitude of the tension in the supporting cable (T):
1. Resolve the weight of the beam into horizontal and vertical components.
Vertical component: W_vertical = Weight of the beam * sin(25°)
Horizontal component: W_horizontal = Weight of the beam * cos(25°)
2. Apply vertical equilibrium:
T + W_vertical + Weight of the load = 0 (since the beam is in equilibrium in the vertical direction)
Solve for T:
T = -W_vertical - Weight of the load
Substitute the values of W_vertical and the weight of the load to find T.
Now, let's find the magnitude of the force exerted on the end of the beam by the bolt (F):
1. Apply horizontal equilibrium:
F + W_horizontal = 0 (since the beam is in equilibrium in the horizontal direction)
Solve for F:
F = -W_horizontal
Substitute the value of W_horizontal to find F.
Calculate the values of T and F using the given data, and you will have the answers in Newtons.