Understanding Forces and Equilibrium: A Comprehensive Analysis
Sum of Non-Concurrent Parallel Forces
The point of application of the two forces differs. When forces are parallel in the same direction, we add the two modules of the two forces: R = F1 + F2. The direction is parallel to the line of the two forces, and their meaning is the same as that of the two forces.
To know the position of point P, at which point your application is now, we can show that the following is met: F1 * x = F2 * (d – x). Where:
- x is the distance separating the point P resulting from the application of one of the forces.
- dx is the distance from the point of application P of the other force.
- d is the separation between the two forces, F1 and F2.
Example
Determine the modulus and point of application of the result of two parallel forces with the same direction, with values of 10N and 15N, separated by a distance of 5m.
First, we represent the forces to better understand the relationship that we apply and results in the results. As forces are parallel in the same direction, the modulus of the result is as follows: R = 15N + 10N = 25N.
If we determine the point P, we apply the following: 10 * x = 15 * (5 – x). Solve the equation for calculating the value of x: 15x = 50 – 10x => 15x + 10x = 50 => 25x = 50 => x = 50/25 = 2m.
Therefore: dx = 5m – 2m = 3m. The resulting force is 2m from the 15N force and 3m from the 10N force.
Opposite Parallel Forces
The result is equal to the difference of the moduli of the two forces: R = F1 – F2. Their direction is a line parallel to the two forces, and their meaning is the same as the larger force. To know the position of point P in which the result has its point of application, one can show that this relation is marked: F1 * x = F2 * (d + x).
Where:
- x is the distance separating point P application resulting from the application point of the larger force.
- d + x is the distance from point P to the point of application of the smaller force.
- d is the separation between the two forces, F1 and F2.
The result is always closer to the component with the larger modulus.
Example
For the same parallel forces in the example above, 10N and 15N, determine the result if they have opposite directions.
As the forces are parallel and in opposite directions, the modulus of the result is the following: R = 15N – 10N = 5N.
To determine the point of application P, use this relationship: 15 * x = 10 * (5 + x). Solve the equation: 15x = 50 + 10x => 15x – 10x = 5x = 50 => x = 50/5 = 10m.
The result is 10 meters from the force of 15N and 15m from the force of 10N, now forming 5N, in the same direction of forces and the same direction as the force of 15N.
Translational Equilibrium
When the sum of the forces acting on a body equals zero.
Rotational Equilibrium
When an object cannot rotate.
Torque
The set of two equal, opposite, and non-concurrent forces.
Moment of a Torque
The moment of a torque is the sum of the products of each pair of forces for the distance from the center. As the products are equal, it is: F = 2 * F * r = F * d. The unit is the Newton-meter (N*m). An object is in rotational equilibrium when it is not rotating because there is no torque or any sum moment on the object in question.
For a body to be in balance, it must be in translational and rotational equilibrium.
Example
On the periphery of the wheel of a car, a torque of 3N is applied. Calculate each moment, knowing that the diameter of the wheel is 40cm.
The torque, expressed in Newton-meters, is: d = 40cm = 0.4m => F = F * d = 3N * 0.4m = 1.2 N*m.
Weight
The weight of a body is the force that attracts it to the Earth. The weight has a vertical direction and points towards the center of the Earth, that is, the vertical relationship between each point. The relationship between weight (P) and mass (m) of a body is: P/m = 9.8 N/kg.
Center of Gravity
The center of gravity is the point of application of the weight of the body.
Balancing of a Body Resting on the Ground
For an object to be stable and not fall into balance, it is necessary that its vertical center of gravity is always on the base of support.
We can distinguish 3 types of balance for a body, depending on whether or not it returns to its stable position after moving from this position:
- Stable balance: When moved, the body returns to its initial stable position.
- Unstable balance: The body does not return to its initial position.
- Indifferent equilibrium: The body never moves from its initial stable position.
Example
Calculate the weight of some items with masses: a) 2kg, b) 0.3g, c) 4mg.
To calculate the weight of each object, first express the mass in SI units, then apply the expression that relates mass with weight: P = m * 9.8 N/kg.
a) m = 2 kg => P = 9.8 * 2 kg * N/kg = 19.6 N