A sign hangs supported by a rod and cable. The cable is at a 30 degree angle to the sign and the force on the rod is totally horizontal. If the sign weighs 200N what is the tension on the cable
To find the tension on the cable, we can break down the forces acting on the sign.
First, let's resolve the weight of the sign into two components: one parallel to the cable and one perpendicular to the cable. The weight of the sign is 200N.
The component parallel to the cable can be found by multiplying the weight of the sign by the cosine of the angle between the weight vector and the cable. In this case, the angle is 30 degrees.
Parallel component = Weight of the sign * cos(angle)
= 200N * cos(30 degrees)
= 200N * 0.866
≈ 173.2N
The perpendicular component can be found by multiplying the weight of the sign by the sine of the angle between the weight vector and the cable. In this case, the angle is 30 degrees.
Perpendicular component = Weight of the sign * sin(angle)
= 200N * sin(30 degrees)
= 200N * 0.5
= 100N
Since the force on the rod is totally horizontal, it means that the vertical components of the forces acting on the sign must balance each other out. This means that the perpendicular component of the weight must be canceled out by the tension in the cable.
Therefore, the tension on the cable is equal to the perpendicular component of the weight, which is 100N.
To find the tension on the cable, we can use trigonometry and decompose the weight of the sign into its vertical and horizontal components.
First, let's find the vertical component of the weight. We can use the formula:
Vertical Component = Weight * sin(angle)
Here, the weight of the sign is given as 200N, and the angle between the cable and the sign is 30 degrees. Plugging these values into the formula, we get:
Vertical Component = 200N * sin(30°) = 100N
Therefore, the vertical component of the weight of the sign is 100N.
Next, let's find the horizontal component of the force. Since the rod is completely horizontal, the horizontal component is equal to the force on the rod. Therefore, the tension on the cable is equal to the horizontal component of the force.
The horizontal component can be found using the formula:
Horizontal Component = Weight * cos(angle)
Plugging in the values we have:
Horizontal Component = 200N * cos(30°) ≈ 173.2N
Therefore, the tension on the cable is approximately 173.2N.