In the general course of physics, two of the simplest types of movement of objects in space are studied - this is translational motion and rotation. If the dynamics of translational motion is based on the use of such quantities as forces and masses, then the concepts of moments are used to quantitatively describe the rotation of bodies. In this article, we will consider by what formula the moment of force is calculated, and for solving what problems this value is used.
Moment of force
Let's imagine a simple system that consists of a material point rotating around an axis at a distance r from it. If a tangential force F, which is perpendicular to the axis of rotation, is applied to this point, then it will lead to the appearance of an angular acceleration of the point. The ability of a force to cause a system to rotate is called torque or moment of force. Calculate according to the following formula:
M¯=[r¯F¯]
In square brackets is the vector product of the radius vector and the force. The radius vector r¯ is a directed segment from the axis of rotation to the point of application of the vector F¯. Taking into account the property of the vector product, for the value of the modulus of the moment, the formula in physics will be written as follows:
M=rFsin(φ)=Fd, where d=rsin(φ).
Here the angle between the vectors r¯ and F¯ is denoted by the Greek letter φ. The value d is called the shoulder of the force. The larger it is, the more torque the force can create. For example, if you open a door by pressing on it near the hinges, then the arm d will be small, so you need to apply more force to turn the door on the hinges.
As you can see from the moment formula, M¯ is a vector. It is directed perpendicular to the plane containing the vectors r¯ and F¯. The direction of M¯ is easy to determine using the right hand rule. To use it, it is necessary to direct four fingers of the right hand along the vector r¯ in the direction of the force F¯. Then the bent thumb will show the direction of the moment of force.
Static torque
The considered value is very important when calculating the equilibrium conditions for a system of bodies with an axis of rotation. There are only two such conditions in statics:
- equality to zero of all external forces that have this or that effect on the system;
- equality to zero of the moments of forces associated with external forces.
Both equilibrium conditions can be mathematically written as follows:
∑i(Fi¯)=0;
∑i(Mi¯)=0.
As you can see, it is the vector sum of quantities that needs to be calculated. As for the moment of force, it is customary to consider its positive direction if the force makes a turn against the clock. Otherwise, a minus sign should be used before the formula for determining the moment.
Note that if the axis of rotation in the system is located on some support, then the corresponding moment reaction force does not create, since its arm is equal to zero.
Moment of force in dynamics
The dynamics of movement of rotation around the axis, like the dynamics of translational movement, has the basic equation, on the basis of which many practical problems are solved. It is called the equation of moments. The corresponding formula is written as:
M=Iα.
In fact, this expression is Newton's second law, if the moment of force is replaced by force, the moment of inertia I - by mass, and the angular acceleration α - by a similar linear characteristic. To better understand this equation, note that the moment of inertia plays the same role as an ordinary mass in translational motion. The moment of inertia depends on the distribution of mass in the system relative to the axis of rotation. The greater the distance of the body to the axis, the greater the value of I.
Angular acceleration α is calculated in radians per second squared. Itcharacterizes the rate of rotation change.
If the moment of force is zero, then the system does not receive any acceleration, which indicates the conservation of its momentum.
Work of moment of force
Since the quantity under study is measured in newtons per meter (Nm), many may think that it can be replaced by a joule (J). However, this is not done because a certain energy quantity is measured in joules, while the moment of force is a power characteristic.
Just like force, moment M can also do work. It is calculated by the following formula:
A=Mθ.
Where the Greek letter θ denotes the angle of rotation in radians, which the system turned as a result of the moment M. Note that as a result of multiplying the moment of force by the angle θ, the units of measurement are preserved, however, the units of work are already used, then Yes, Joules.
