A weight (with a mass of 42 kg) is suspended
from a point near the right-hand end of a uni-
form boom with a mass of 37 kg
.
To support
the uniform boom a cable runs from this same
point to a wall (the left-hand vertical coordi-
nate in the figure) and by a pivot on the same
wall at an elevation of 4 m
.
To find the tension in the cable, we can use the principle of torque equilibrium.
First, we need to calculate the torque caused by the weight of the boom. The torque is the product of the force and the perpendicular distance from the point of rotation (pivot) to the line of action of the force.
The force due to the weight of the boom can be calculated as the mass of the boom multiplied by the acceleration due to gravity. So, the force due to the weight of the boom is given by:
Force_boom = mass_boom * acceleration_due_to_gravity
Plugging in the values, we have:
Force_boom = 37 kg * 9.8 m/s^2 = 362.6 N
Next, we need to calculate the perpendicular distance from the pivot to the line of action of the force. In this case, the vertical distance from the pivot to the center of mass of the boom is given as 4 m.
Now, we can calculate the torque caused by the weight of the boom:
Torque_boom = Force_boom * distance_boom
Torque_boom = 362.6 N * 4 m = 1450.4 Nm
To maintain torque equilibrium, the tension in the cable must exert an equal and opposite torque. The pivot on the wall acts as a fixed fulcrum, so the torque caused by the tension in the cable will be clockwise.
Let's assume the tension in the cable is T. The perpendicular distance from the pivot to the line of action of the tension (which is the length of the boom) is unknown.
Now, we can write the equation for torque equilibrium:
Torque_cable = Torque_boom
T * distance_cable = 1450.4 Nm
Solving for T, we have:
T = 1450.4 Nm / distance_cable
So, to find the tension in the cable, we need to know the distance from the pivot to the point where the cable is attached to the boom.