Often in physics, it is necessary to solve the problem of calculating the equilibrium in complex systems that have many effective forces, levers and axes of rotation. In this case, it is easiest to use the concept of moment of force. This article provides all the necessary formulas with detailed explanations that should be used to solve problems of this type.
What will it be about?
Many people probably noticed that if you act with any force on an object fixed at some point, then it starts to rotate. A striking example is the door to a house or room. If you take it by the handle and push it (apply force), then it will begin to open (turn on hinges). This process is a manifestation in everyday life of the action of a physical quantity, which is called the moment of force.
From the described door example it follows that the quantity in question indicates the ability of a force to rotate, which is its physical meaning. Also, this value is called the torsion moment.
Determination of the moment of force
Before defining the value under consideration, we give a simple drawing.
So, the figure shows the lever (blue), which is fixed on the axis (green). This lever has a length d, and a force F is applied to its end. What will happen to the system in this case? True, the lever will begin to rotate counterclockwise, if you look at it from above (note that if you strain your imagination a bit and imagine that your gaze is directed from below to the lever, then it will rotate clockwise).
Let the axis attachment point be called O, and the point of force application - P. Then, we can write the following mathematical expression:
OP ¯ * F ¯ = M ¯ F O.
Where OP ¯ is the vector that is directed from the axis to the end of the lever, it is also called the lever of force, F ¯ is the vector of the applied force to the point P, and M ¯F O is the moment of force relative to the point O (axis). This formula is a mathematical definition of the physical quantity in question.
Direction of torque and rule of the right hand
The expression above is a vector product. As you know, its result is also a vector that is perpendicular to the plane passing through the corresponding factor vectors. This condition is satisfied by two directions of the quantity M ¯ F O (down and up).
To uniquely identify it, you should use the so-called right-hand rule. It can be formulated as follows: if four fingers of the right hand are bent into a half-arc and this half-arc is directed so that it runs along the first vector (the first factor in the formula) and goes to the end of the second, then the thumb protruding up will indicate the direction of the torsion moment. We also note that before using this rule, it is necessary to set the multiplied vectors so that they exit from one point (their origin must coincide).
In the case of the figure in the previous paragraph, we can say, applying the rule of the right hand, that the moment of force relative to the axis will be directed upwards, that is, towards us.
In addition to the noted method for determining the direction of the vector M ¯ F O , there are two more. We give them:
- The torsion moment will be directed in such a way that if you look at the rotating lever from the end of its vector, then the latter will move against the clock hands. It is generally accepted that this direction of the moment is positive in solving various problems.
- If you twist the gimlet clockwise, then the torsion moment will be directed towards the movement (deepening) of the gimlet.
All the above definitions are equivalent, so everyone can choose the one that is convenient for him.
So, it was found that the direction of the moment of force is parallel to the axis around which the corresponding lever rotates.
Angled force
Consider the figure below.
Here we also see a lever of length L, fixed at a point (indicated by an arrow). The force F acts on it, however, it is directed at a certain angle Φ (phi) to the horizontal lever. The direction of the moment M ¯F O in this case will be the same as in the previous figure (towards us). To calculate the absolute value or modulus of this quantity, it is necessary to use the property of the vector product. According to it, for the considered example, we can write the expression: M F O = L * F * sin (180 o -Φ) or, using the sine property, we rewrite:
M F O = L * F * sin (Φ).
The figure also shows the completed rectangular triangle, the sides of which are the lever itself (hypotenuse), the line of action of the force (leg) and the side of length d (second leg). Given that sin (Φ) = d / L, this formula takes the form: M F O = d * F. It can be seen that the distance d is the distance from the point of attachment of the lever to the line of action of the force, that is, d is the lever of force.
Both formulas considered in this clause, which follow directly from the definition of the torsion moment, are useful in solving practical problems.
Torsion torque units
Using the definition, it can be established that the value of M F O should be measured in newtons per meter (N * m). Indeed, in the form of these units it is also used in SI.
Note that N * m is a unit of work that is expressed in joules, like energy. Nevertheless, joules are not used for the concept of the moment of force, since this value reflects precisely the possibility of realizing the latter. However, there is a connection with the unit of work: if, as a result of the action of the force F, the lever is completely rotated around its rotation point O, then the perfect work will be equal to A = M F O * 2 * pi (2 * pi is the angle in radians, which corresponds to 360 o ) In this case, the unit of measure of the moment M F O can be expressed in joules per radian (J / rad.). The latter, along with N * m, is also used in the SI system.
Varignon's theorem
At the end of the 17th century, the French mathematician Pierre Varignon, studying the equilibrium of systems with levers, first formulated a theorem that now bears his last name. It is formulated as follows: the total moment of several forces is equal to the moment of the resulting one force, which is applied to a point relative to the same axis of rotation. Mathematically, it can be written as follows:
M ¯ 1 + M ¯ 2 + ... + M ¯ n = M ¯ = d ¯ * ∑ n i = 1 (F ¯ i ) = d ¯ * F ¯ .
This theorem can be conveniently used to calculate the torsion moments in systems with several acting forces.
Next, we give an example of using the above formulas to solve problems in physics.
Wrench task
One striking example of demonstrating the importance of considering the moment of force is the process of loosening the nuts with a wrench. To unscrew the nut, you need to apply some torsion moment. It is necessary to calculate what force should be applied at point A in order to start unscrewing the nut if this force at point B is 300 N (see the figure below).
Two important things follow from the given figure: firstly, the distance OB is twice as large as OA; secondly, the forces F A and F B are directed perpendicular to the corresponding lever with the axis of rotation coinciding with the center of the nut (point O).
The torsion moment for this case can be written in scalar form as follows: M = OB * F B = OA * F A. Since OB / OA = 2, then this equality will be fulfilled only when F A will be 2 times greater than F B. From the conditions of the problem we obtain that F A = 2 * 300 = 600 N. That is, the longer the key length, the easier it is to unscrew the nut.
The problem with two balls of different masses
The figure below shows a system that is in equilibrium. It is necessary to find the position of the fulcrum if the length of the board is 3 meters.
Since the system is in equilibrium, the sum of the moments of all forces is zero. Three forces act on the board (the weights of two balls and the reaction force of the support). Since the support force does not create a torsion moment (the length of the lever is zero), only two moments remain, created by the weight of the balls.
Let the equilibrium point be at a distance x from the edge where the ball weighing 100 kg lies. Then we can write the equation: M 1 -M 2 = 0. Since the body weight is determined by the formula m * g, then we have: m 1 * g * x - m 2 * g * (3-x) = 0. Reduce g and substitute the data, we get: 100 * x - 5 * (3-x) = 0 => x = 15/105 = 0.143 m or 14.3 cm.
Thus, in order for the system to be in equilibrium, it is necessary to establish a reference point at a distance of 14.3 cm from the edge where a ball weighing 100 kg will lie.