For an object in equilibrium the sum of the torques acting on it vanishes only if each torque is calculated based on your:
This makes not much sense. The sum does not vanish, it is zero. Zero is not vanishing.
The sum of all torques about ANY point is zero in a system in equilibium.
Now, for the last"is calculated based on your:" I haven't a clue what the question is asking.
To calculate the sum of torques acting on an object in equilibrium, you need to consider the torques produced by each individual force acting on the object. The torque produced by a force is determined by multiplying the magnitude of the force by the perpendicular distance from the point of rotation, known as the axis of rotation, to the line of action of the force.
To find the torque produced by each force, you can follow these steps:
1. Identify all the external forces acting on the object. These forces can include gravitational force, applied forces, or any other forces in the system.
2. Determine the line of action for each force. The line of action is an imaginary line that represents the direction of the force.
3. Identify the perpendicular distance, often denoted as 'r', between the axis of rotation and the line of action for each force. It is the shortest distance from the axis of rotation to the line of action of the force.
4. Calculate the torque produced by each force using the equation: torque = force x perpendicular distance (τ = F x r). Ensure that the force and the distance are in the correct units. The SI unit for torque is Newton-meter (N·m).
5. Finally, sum up all the torques produced by each force. If the object is in equilibrium, the sum of all the torques should be zero. Mathematically, this can be expressed as Στ = 0, where Στ represents the sum of torques.
Remember that torques are vector quantities, meaning they have both magnitude and direction. So, when summing the torques, pay attention to the direction as well.