A weight (with a mass of 71 kg) is suspended
from a point near the right-hand end of a uni-
form boom with a mass of 51 kg . To support
the uniform boom a cable runs from this same
point to a wall Vertical Height (m) is 7
Horizontal Distance (m) 6
and by a pivot on the same
wall at an elevation of 3 m.
Calculate the tension T in the cable. The
acceleration of gravity is 9.8 m/s2 .
Answer in units of N
To calculate the tension in the cable, we can use the principle of torque balance. Torque is the rotational analog of force, and it depends on the force applied, the distance from the pivot point, and the angle between the force and the lever arm.
First, let's take a look at the forces acting on the boom. There are three forces: the weight of the boom acting downward, the tension in the cable pulling in a diagonal direction, and the pivot force acting upward.
1. The weight of the boom can be calculated by multiplying its mass (51 kg) by the acceleration due to gravity (9.8 m/s^2). So the weight of the boom is 499.8 N (Newtons).
2. Next, let's find the torque caused by the weight of the boom. The torque is defined as the force multiplied by the perpendicular distance from the pivot point. Since the weight of the boom is acting at its center, the perpendicular distance is half its length, which is 3 m. So the torque caused by the weight is 499.8 N * 3 m = 1499.4 N·m.
3. The tension in the cable can be broken down into vertical and horizontal components. The vertical component of tension is balanced by the pivot force, so we can ignore it for our torque calculation.
4. To find the horizontal component of tension, we need to find the horizontal distance from the pivot to the suspension point. The horizontal distance is given as 6 m.
5. Now we can find the torque caused by the tension in the cable. The torque caused by the tension is equal to the tension multiplied by the horizontal distance. Let's call the tension T. So the torque caused by the tension is T * 6 m.
6. Lastly, since the boom is in equilibrium (not rotating), the total torque must be zero. This means the torque caused by the weight of the boom is equal and opposite to the torque caused by the tension in the cable. Therefore, we can set up the equation:
1499.4 N·m = T * 6 m
Now we can solve for T. Dividing both sides of the equation by 6 m, we get:
T = 1499.4 N·m / 6 m
T ≈ 249.9 N
Therefore, the tension in the cable is approximately 249.9 N.