Consider a lever rod of length 7.96 m, weight 71 N and uniform density. The lever rod is pivoted on one end and is supported by a cable attached at a point 1.74 m from the
other end:

The lever rod is in equilibrium at angle of 54◦ from the vertical wall. The cable makes angle of 61◦ with the rod.
What is the tension of the supporting ca- ble?
Answer in units of N.
To find the tension of the supporting cable, we can use the principle of moments and resolve forces in the horizontal and vertical directions.
Step 1: Resolve forces vertically:
Taking the anticlockwise direction as positive, the weight of the rod can be decomposed into vertical and horizontal components:
Vertical Component = Weight * sin(54°)
Vertical Component = 71 N * sin(54°)
Vertical Component ≈ 56.52 N
Step 2: Resolve forces horizontally:
Taking the clockwise direction as positive, the tension in the cable can be decomposed into horizontal and vertical components:
Horizontal Component = Tension * cos(61°)
Horizontal Component = Tension * cos(61°)
Step 3: Apply the principle of moments:
The principle of moments states that the sum of the moments about any point must be zero for an object in equilibrium.
Let's take the point where the rod is pivoted as the reference point.
Using the principle of moments about the pivot point:
Clockwise moment = Anti-clockwise moment
(horizontal component of tension) * (distance from the pivot to the cable) = (vertical component of weight) * (distance from the pivot to the weight)
(Tension * cos(61°)) * 1.74 m = 56.52 N * 5.22 m
Tension * cos(61°) ≈ (56.52 N * 5.22 m) / 1.74 m
Tension ≈ (56.52 N * 5.22 m / 1.74 m) / cos(61°)
Step 4: Calculate the tension:
Tension ≈ 168.67 N
Therefore, the tension of the supporting cable is approximately 168.67 N.
To find the tension of the supporting cable, we need to analyze the forces acting on the lever rod and use the conditions for equilibrium. In this case, since the lever rod is in equilibrium and not moving, the sum of the forces acting on it must be zero.
Let's break down the forces acting on the lever rod:
1. Weight of the rod (acting vertically downwards): This force can be calculated using the formula: weight = mass * gravitational acceleration. The weight is given as 71 N.
2. Tension in the supporting cable: This force is acting at an angle of 61 degrees with the rod. We need to find the vertical and horizontal components of this tension force.
To find the vertical component, we can use the formula: vertical component = tension * sin(angle). Here, angle is given as 61 degrees.
To find the horizontal component, we can use the formula: horizontal component = tension * cos(angle).
Now, let's consider the torque or moment equation about the pivot point of the lever rod:
The clockwise torque caused by the weight of the rod can be calculated as the product of the weight and the perpendicular distance from the pivot point to the line of action of the weight force.
The anticlockwise torque caused by the tension in the cable can be calculated as the product of the vertical component of tension and the perpendicular distance from the pivot point to the line of action of the tension force.
Since the rod is in equilibrium, these torques must be equal: clockwise torque = anticlockwise torque.
The torque due to weight is given by: weight * distance from the pivot point to the line of action of the weight force * sin(angle between weight force and lever arm).
The torque due to tension is given by: vertical component of tension * distance from the pivot point to the line of action of the tension force.
Setting up the equation using the given values and solving for tension will give us the desired answer.
Let me calculate it for you.