A display sign is supported by a guy making an angle of 37o with the horizontal and by a bracket at the upper end near the vertical wall. The sign weighs 150N. Find
A. The tension in wire
B. The force exerted by the bracket on the sign.
To find the tension in the wire and the force exerted by the bracket on the sign, we can break down the forces acting on the sign.
Let's start by drawing a diagram to visualize the situation. The guy wire makes an angle of 37 degrees with the horizontal, and the sign is supported by the bracket on the upper end near the vertical wall.
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Guy Wall
Now, let's consider the forces acting on the sign. There are two forces involved:
1. Tension in the guy wire: This force is pulling the sign inwards and is responsible for balancing the weight of the sign.
2. Force exerted by the bracket: This force is perpendicular to the wall and keeps the sign from falling.
Let's assume that the tension in the wire is T and the force exerted by the bracket is F.
Let's break down the horizontal and vertical components of the forces:
Horizontal forces:
- Tension in the wire (T) has no horizontal component.
- The force exerted by the bracket (F) has no horizontal component.
Therefore, the horizontal forces are balanced.
Vertical forces:
- Tension in the wire (T) has a vertical component, which helps to balance the weight of the sign.
- The force exerted by the bracket (F) has no vertical component.
Since the sign weighs 150N, the vertical force exerted by the weight of the sign is 150N.
Using trigonometry, we can determine the equation for the vertical component of the tension in the wire:
Vertical component of T = T * sin(37)
According to Newton's second law of motion, for the sign to be in equilibrium, the sum of the vertical forces should be zero:
Sum of vertical forces = vertical component of T - weight of the sign + Force exerted by the bracket = 0
150N = T * sin(37) + F
Now we have two equations with two unknowns:
1. Vertical component of T = T * sin(37)
2. 150N = T * sin(37) + F
We can solve these equations simultaneously to find the values of T and F.
To find the tension in the wire and the force exerted by the bracket on the sign, we can use trigonometry and resolve the forces acting on the sign.
Let's break down the forces acting on the sign:
1. Weight of the sign (150N): This force acts vertically downward.
2. Tension in the wire: This force acts at an angle of 37 degrees with the horizontal.
3. Force exerted by the bracket: This force acts vertically upward.
Now, let's calculate the tension in the wire first.
Step 1: Resolve the weight of the sign into horizontal and vertical components:
- Vertical component = Weight * sin(angle)
Vertical component = 150N * sin(37o)
Vertical component ≈ 90.16N
Step 2: Calculate the tension in the wire:
- The vertical component of the tension balances the vertical component of the weight:
Tension * cos(angle) = Vertical component
Tension * cos(37o) = 90.16N
Tension ≈ 90.16N / cos(37o)
Tension ≈ 113.76N
Therefore, the tension in the wire is approximately 113.76N.
Next, let's find the force exerted by the bracket on the sign.
Step 1: Resolve the weight of the sign into horizontal and vertical components:
- Horizontal component = Weight * cos(angle)
Horizontal component = 150N * cos(37o)
Horizontal component ≈ 119.76N
Step 2: Calculate the force exerted by the bracket:
- The horizontal component of the force exerted by the bracket balances the horizontal component of the weight:
Force exerted by bracket = Horizontal component
Force exerted by bracket ≈ 119.76N
Therefore, the force exerted by the bracket on the sign is approximately 119.76N.
In summary:
A. The tension in the wire is approximately 113.76N.
B. The force exerted by the bracket on the sign is approximately 119.76N.