The moment represents the rotational effect produced when the force acts on the objectphysical quantity。The product of force and force arm is called force pairRotating shaftTorque.[7]That is, the magnitude of the moment of a force on a point is induced by the action line from the point to the forceverticalLength (i.eLever arm）Multiplied by the force, its direction is perpendicular to the plane formed by the vertical line and the forceRight-Hand Rule To determine.
The magnitude of the moment of force on an axis is equal to the projection of the moment of force on any point on the axis on the axis.
Moment can make objects obtainangular accelerationAnd can change the momentum moment of the object. For the same object, the greater the torque, the easier it is to change the rotating state.[1]
Moment of force The physical quantity that the force exerts a rotational effect on an object.It can be divided into moment of force to axis and moment of force to point.That is:M=r×F。amongrIs the position vector from the axis of rotation to the point of force,FIs a vector force;The moment is alsovector。[1-2]
Moment of force to point
Moment is a physical quantity that measures the rotational effect of force on an object.It can be divided into moment of force to point and moment of force to axis.The moment of a force on a point is a physical quantity that measures the effect of the force on an object rotating around the point.The moment of force F to a point O is defined as: the vector diameter r of point A of force F relative to point O and the vector product of force F is Mzero(F) Means, Mzero(F) =r × F, the moment of the force to the point is a vector, and the magnitude is equal to the product of the magnitude of F and the vertical distance d (called the arm of force) of the action line from point O to F, or equal to the value ofparallelogramArea rFsin α, α is the angle between r and F.Mzero(F) The direction is perpendicular to the plane formed by r and F. r, F and M. (F) meet the right-hand spiral relationship.The moment of force to point can be defined for any point in space.Since the moment of a force to a point depends on the position vector radius r of the point of action of the force, the moment of the same force to different points in space is different.When the action line of a force crosses a point in space, the moment of the force at that point is zero.If there are several common forces (acting point is A) Fi (i=1, 2,..., n) acting on the object, the resultant force F=Fone+Ftwo+…+Fn, then the moment M of the resultant force to point Ozero(F)=r×(Fone+Ftwo+……+F)=r×Fone+r×Ftwo+…+r×Fn=M01+M02…+M0nThat is, the torque of the resultant force to a point O is equal to the vector sum of the torque of each component force to the same point.Vector Mzero(F) It is called the principal moment of this force system to point O.[2]
Moment of force to axis
The moment of a force on an axis is a physical quantity that measures the rotational effect of the force on an object around the axis.It is defined as the projection of the moment M of force F to point O on any axis OZ axis passing through point O is called the moment of force F to OZ axis, expressed in Mz, Mz=Mcos β, β is the included angle between the positive direction of vector M and OZ axis, and it is specified that the positive direction of object rotation and the positive direction of OZ axis meet the right-hand spiral relationship, as shown by the arrow in Figure 2.Mz is an algebraic quantity, its positive and negative represent the rotation tendency of the object, Mz>0 means that the force F makes the rotation tendency of the object consistent with the positive direction of rotation, and Mz<0 is the opposite.It must be pointed out that the moment of force F on different points of OZ axis is different, but the projection of these moments on OZ axis is equal.Therefore, it can be said that the projection of force F on OZ axis of any moment on OZ axis is equal to the moment of force F on OZ axis.If the force F is parallel to the OZ axis or the action line of F intersects the OZ axis, the moment of F on the OZ axis is zero.The moment of force F to the OZ axis can also be defined as the projection of force F in the plane perpendicular to the OZ axis, the projection of the moment of force F on the OZ axis of the intersection point of the plane and the OZ axis, the projection of the moment of force F on the OZ axis: [Moz (F)]z=[M0(F⊥)]z=[r×F⊥]z。When the direction of Moz (F) is consistent with the positive direction of OZ axis, it is positive, which means that the force F has a tendency to make the object rotate counterclockwise when observing the arrow facing OZ axis, otherwise it is the opposite.Or the direction of Moz (F) and the rotation tendency of the object meet the right-hand spiral relationship.The moment of force to axis can be defined for any axis of space.The unit of moment is Newton meter (N · m).[2]
Figure 2
nature
Announce
edit
1. ForceFThe moment of point O is not only determined by the force, but also related to the position of the moment center.The moment varies with the position of the moment center.[3]
2. When the force is zero orLever armWhen it is zero, the moment is zero.[3]
3. When the force moves along its line of action, the torque does not change because the size, direction and arm of the force do not change.[3]
4. The algebraic sum of the moments of two mutually balanced forces on the same point is equal to zero.[3]
application
Announce
edit
When tightening nuts with a wrench in life, the force F acting on the wrench makes the wrench rotate around point O. The larger the force F used in the hand, the tighter the nut.This shows that the rotation effect of the wrench around the fulcrum O is not only proportional to the force F, but also to the vertical distance r (calledLever arm）It is also proportional."Moment" is used to measure the effect of force on the rotation of an object around a fulcrum (called the center of moment).The moment of force F to point O of the moment center is referred to as moment, which is expressed by M (F). Its size is equal to the product of force F and force arm r, that is, M (F)=F · r, as shown in Figure 3.[4]
Figure 3
Related concepts
Announce
edit
Moment of rigid body
If acting onrigid bodyThe external force on is in the plane perpendicular to the rotation axis, as shown in Figure 4 (a), then the moment M of the external force F on the rotation axis is M=r × F.The size of M is M=Frsin θ=Fd;The direction of M is perpendicular to the plane formed by r and FRight hand spiral ruleConfirm that the direction of moment M is along the rotation axis during the rotation of fixed axis.If the external force acting on the rigid body is not in the plane perpendicular to the rotation axis, as shown in Figure 4 (b).Because the external force parallel to the rotation axis does not play a role in the rotation of the rigid body around the axis during the rotation of a fixed axis, the force F is divided into two parts in the planevectorOnly then has an effect on rigid body rotation.The force F is decomposed into the component F parallel to the shaft and the component F perpendicular to the shaft⊥Only the component F can make the rigid body rotate, then the torque can be written as M=r × F⊥In the fixed axis rotation, if the force F passes through the rotation axis, the moment M is equal to zero, and the rigid body cannot rotate;If several external forces act on a rigid body rotating around a fixed axis at the same time, and these external forces are in a plane perpendicular to the rotation axis, their combined external moments are equal to the algebraic sum of these external moments.If there is an interaction force (internal force) between particles in the rigid body, the force between particles always occurs in pairs, andNewton's Third LawTherefore, when discussing the fixed axis rotation of the rigid body, the combined internal moment of these internal forces on the rotation axis is zero.[5]
Figure 4
Rotation law of fixed axis rotation
The change of the motion state of the rigid body when it rotates with a fixed axis depends on the external torque M applied to the rigid body.Just as the resultant force on a particle is the cause of acceleration a, M is the cause of angular acceleration a.When the external torque is givenMoment of inertiaLarge, the obtainedangular accelerationSmall, that is, the angular velocity changes slowly, that is, the inertia of maintaining the original rotating state is large;On the contrary, if the rotational inertia of the rigid body is small, the angular acceleration obtained will be large, that is, the angular velocity will change quickly, that is, the inertia of the original rotational state will be small.The law of rotation is a quantitative formula of dynamics of rigid body rotating with fixed axis, and it is a system of particlesAngular momentum theoremThe special form of rigid body rotating with fixed axis is also the instantaneous law of rigid body rotating with fixed axis.If the moment corresponds to the force, the moment of inertia corresponds to the mass, and the angular acceleration corresponds to the acceleration, it is obvious that the law of rotation andNewton's second lawIts form is similar to that of Newton's second law in particle dynamics.[5]
