The moment in the event that. Moment of inertia for dummies: definition, formulas, examples of problem solving

The rule of leverage, discovered by Archimedes in the third century BC, existed for almost two thousand years, until in the seventeenth century with light hand the French scientist Varignon did not receive a more general form.

Torque rule

The concept of torque was introduced. The moment of force is physical quantity, equal to the product of the force by its shoulder:

where M is the moment of force,
F - strength,
l - leverage of force.

From the lever equilibrium rule directly The rule of moments of forces follows:

F1 / F2 = l2 / l1 or, by the property of proportion, F1 * l1= F2 * l2, that is, M1 = M2

In verbal expression, the rule of moments of forces is as follows: a lever is in equilibrium under the action of two forces if the moment of the force rotating it clockwise is equal to the moment of the force rotating it counterclockwise. The rule of moments of force is valid for any body fixed around a fixed axis. In practice, the moment of force is found as follows: in the direction of action of the force, a line of action of the force is drawn. Then, from the point at which the axis of rotation is located, a perpendicular is drawn to the line of action of the force. The length of this perpendicular will be equal to the arm of the force. By multiplying the value of the force modulus by its arm, we obtain the value of the moment of force relative to the axis of rotation. That is, we see that the moment of force characterizes the rotating action of the force. The effect of a force depends on both the force itself and its leverage.

Application of the rule of moments of forces in various situations

This implies the application of the rule of moments of forces in different situations. For example, if we open a door, then we will push it in the area of ​​the handle, that is, away from the hinges. You can do a basic experiment and make sure that pushing the door is easier the further you apply force from the axis of rotation. Practical experiment in in this case is directly confirmed by the formula. Since, in order for the moments of forces at different arms to be equal, it is necessary that the larger arm correspond to a smaller force and, conversely, the smaller arm correspond to a larger one. The closer to the axis of rotation we apply the force, the greater it should be. The farther from the axis we operate the lever, rotating the body, the less force we will need to apply. Numeric values are easily found from the formula for the moment rule.

It is precisely based on the rule of moments of force that we take a crowbar or a long stick if we need to lift something heavy, and, having slipped one end under the load, we pull the crowbar near the other end. For the same reason, we screw in the screws with a long-handled screwdriver, and tighten the nuts with a long wrench.

We often hear the expressions: “it is inert”, “move by inertia”, “moment of inertia”. In a figurative sense, the word “inertia” can be interpreted as a lack of initiative and action. We are interested in the direct meaning.

What is inertia

According to definition inertia in physics, it is the ability of bodies to maintain a state of rest or motion in the absence of external forces.

If everything is clear with the very concept of inertia on an intuitive level, then moment of inertia– a separate question. Agree, it is difficult to imagine in your mind what it is. In this article you will learn how to solve basic problems on the topic "Moment of inertia".

Determination of moment of inertia

From school course it is known that mass – a measure of the inertia of a body. If we push two carts of different masses, then the heavier one will be more difficult to stop. That is, the greater the mass, the greater external influence necessary to change body movement. What is considered applies to translational motion, when the cart from the example moves in a straight line.

By analogy with mass and translational motion, the moment of inertia is a measure of the inertia of a body at rotational movement around the axis.

Moment of inertia– a scalar physical quantity, a measure of the inertia of a body during rotation around an axis. Denoted by the letter J and in the system SI measured in kilograms times a square meter.

How to calculate the moment of inertia? Eat general formula, which is used in physics to calculate the moment of inertia of any body. If a body is broken into infinitesimal pieces with a mass dm , then the moment of inertia will be equal to the sum the products of these elementary masses by the square of the distance to the axis of rotation.

This is the general formula for moment of inertia in physics. For a material point of mass m , rotating around an axis at a distance r from her, this formula takes the form:

Steiner's theorem

What does the moment of inertia depend on? From mass, position of the axis of rotation, shape and size of the body.

The Huygens-Steiner theorem is a very important theorem that is often used in solving problems.

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The Huygens-Steiner theorem states:

The moment of inertia of a body relative to an arbitrary axis is equal to the sum of the moment of inertia of the body relative to an axis passing through the center of mass parallel to an arbitrary axis and the product of the body mass by the square of the distance between the axes.

For those who do not want to constantly integrate when solving problems of finding the moment of inertia, we present a drawing indicating the moments of inertia of some homogeneous bodies that are often encountered in problems:

An example of solving a problem to find the moment of inertia

Let's look at two examples. The first task is to find the moment of inertia. The second task is to use the Huygens-Steiner theorem.

Problem 1. Find the moment of inertia of a homogeneous disk of mass m and radius R. The axis of rotation passes through the center of the disk.

Solution:

Let us divide the disk into infinitely thin rings, the radius of which varies from 0 before R and consider one such ring. Let its radius be r, and mass – dm. Then the moment of inertia of the ring is:

The mass of the ring can be represented as:

Here dz– height of the ring. Let's substitute the mass into the formula for the moment of inertia and integrate:

The result was a formula for the moment of inertia of an absolute thin disk or cylinder.

Problem 2. Let again there be a disk of mass m and radius R. Now we need to find the moment of inertia of the disk relative to the axis passing through the middle of one of its radii.

Solution:

The moment of inertia of the disk relative to the axis passing through the center of mass is known from the previous problem. Let's apply Steiner's theorem and find:

By the way, on our blog you can find other useful materials on physics and.

We hope that you will find something useful for yourself in the article. If difficulties arise in the process of calculating the inertia tensor, do not forget about the student service. Our specialists will advise on any issue and help solve the problem in a matter of minutes.

Definition 1

The moment of force is represented by a torque or torque, being at the same time a vector physical quantity.

It is defined as the vector product of the force vector, as well as the radius vector, which is drawn from the axis of rotation to the point of application of the specified force.

The moment of force is a characteristic of the rotational effect of a force on a solid body. The concepts of “rotating” and “torque” moments will not be considered identical, since in technology the concept of “rotating” moment is considered as an external force applied to an object.

At the same time, the concept of “torque” is considered in the format of internal force that arises in an object under the influence of certain applied loads (a similar concept is used for the resistance of materials).

Concept of moment of force

The moment of force in physics can be considered in the form of the so-called “rotational force”. The SI unit of measurement is the newton meter. The moment of a force may also be called the "moment of a couple of forces", as noted in Archimedes' work on levers.

Note 1

IN simple examples, when a force is applied to the lever in a perpendicular relation to it, the moment of force will be determined as the product of the magnitude of the specified force and the distance to the axis of rotation of the lever.

For example, a force of three newtons applied at a distance of two meters from the axis of rotation of the lever creates a moment equivalent to a force of one newton applied at a distance of 6 meters to the lever. More precisely, the moment of force of a particle is determined in the vector product format:

$\vec (M)=\vec(r)\vec(F)$, where:

• $\vec (F)$ represents the force acting on the particle,
• $\vec (r)$ is the radius of the particle vector.

In physics, energy should be understood as a scalar quantity, while torque would be considered a (pseudo) vector quantity. The coincidence of the dimensions of such quantities will not be accidental: a moment of force of 1 N m, which is applied through a whole revolution, performing mechanical work, imparts energy of 2 $\pi$ joules. Mathematically it looks like this:

$E = M\theta$, where:

• $E$ represents energy;
• $M$ is considered to be the torque;
• $\theta$ will be the angle in radians.

Today, the measurement of moment of force is carried out by using special load sensors of strain gauge, optical and inductive types.

Formulas for calculating moment of force

An interesting thing in physics is the calculation of the moment of force in a field, produced according to the formula:

$\vec(M) = \vec(M_1)\vec(F)$, where:

• $\vec(M_1)$ is considered the lever moment;
• $\vec(F)$ represents the magnitude of the acting force.

The disadvantage of such a representation is the fact that it does not determine the direction of the moment of force, but only its magnitude. If the force is perpendicular to the vector $\vec(r)$, the moment of the lever will be equal to the distance from the center to the point of the applied force. In this case, the moment of force will be maximum:

$\vec(T)=\vec(r)\vec(F)$

When a force performs a certain action at any distance, it will perform mechanical work. In the same way, the moment of force (when performing an action through an angular distance) will do work.

$P = \vec (M)\omega $

In the existing international system measurements, power $P$ will be measured in Watts, and the moment of force itself will be measured in Newton meters. Wherein angular velocity is defined in radians per second.

Moment of several forces

Note 2

When a body is exposed to two equal and also oppositely directed forces that do not lie on the same straight line, the body is not in a state of equilibrium. This is explained by the fact that the resulting moment of the indicated forces relative to any of the axes does not have a zero value, since both represented forces have moments directed in the same direction (a pair of forces).

In a situation where the body is fixed on an axis, it will rotate under the influence of a couple of forces. If a pair of forces is applied to a free body, it will then begin to rotate around an axis passing through the center of gravity of the body.

The moment of a pair of forces is considered to be the same with respect to any axis that is perpendicular to the plane of the pair. In this case, the total moment $M$ of the pair will always be equal to the product of one of the forces $F$ and the distance $l$ between the forces (arm of the pair), regardless of the types of segments into which it divides the position of the axis.

$M=(FL_1+FL-2) = F(L_1+L_2)=FL$

In a situation where the resultant moment of several forces is equal to zero, it will be considered the same relative to all axes parallel to each other. For this reason, the effect on the body of all these forces can be replaced by the action of just one pair of forces with the same moment.

Moment of power (synonyms: torque, torque, torque, torque) - vector physical quantity equal to the vector product of the radius vector drawn from the axis of rotation to the point of application of the force and the vector of this force. Characterizes the rotational action of a force on a solid body.

The concepts of “rotating” and “torque” moments in general case are not identical, since in technology the concept of “rotating” moment is considered as external force applied to an object, and “torque” is an internal force that arises in an object under the influence of applied loads (this concept is used in the resistance of materials).

General information

Special cases

Lever torque formula

Very interesting a special case, represented as a definition of the moment of force in the field:

$\backslash left|\backslash vec\; M\backslash right|\; =\; \backslash left|\backslash vec(M)\_1\backslash right|\; \backslash left|\backslash vec\; F\backslash right|$, Where: $\backslash left|\backslash vec(M)\_1\backslash right|$- lever moment, $\backslash left|\backslash vec\; F\backslash right|$- the magnitude of the acting force.

The problem with this representation is that it does not give the direction of the moment of force, but only its magnitude. If the force is perpendicular to the vector $\backslash vec\; r$, the moment of the lever will be equal to the distance to the center and the moment of force will be maximum:

$\backslash left|\backslash vec(T)\backslash right|\; =\; \backslash left|\backslash vec\; r\backslash right|\; \backslash left|\backslash vec\; F\backslash right|$

Force at an angle

If strength $\backslash vec\; F$ directed at an angle $\backslash theta$ to lever r, then $M\; =\; r\; F\backslash sin\backslash theta$.

Static balance

In order for an object to be in equilibrium, not only the sum of all forces must be zero, but also the sum of all moments of force around any point. For a two-dimensional case with horizontal and vertical forces: the sum of forces in two dimensions ΣH=0, ΣV=0 and the moment of force in the third dimension ΣM=0.

Moment of force as a function of time

$\backslash vec\; M\; =\; \backslash frac(d\backslash vec\; L)(dt)$,

Where $\backslash vec\; L$- moment of impulse.

Let's take a solid body. Movement solid can be represented as the movement of a specific point and rotation around it.

The angular momentum relative to point O of a rigid body can be described through the product of the moment of inertia and angular velocity relative to the center of mass and the linear motion of the center of mass.

$\backslash vec(L\_o)\; =\; I\_c\backslash ,\backslash vec\backslash omega\; +$

We will consider rotating movements in the Koenig coordinate system, since it is much more difficult to describe the motion of a rigid body in the world coordinate system.

Let's differentiate this expression with respect to time. And if $I$ is a constant value in time, then

$\backslash vec\; M\; =\; I\backslash frac(d\backslash vec\backslash omega)(dt)\; =\; I\backslash vec\backslash alpha$,

Relationship between torque and work

$A\; =\; \backslash int\_(\backslash theta\_1)^(\backslash theta\_2)\; \backslash left|\backslash vec\; M\backslash right|\; \backslash mathrm(d)\backslash theta$

In the case of constant torque we get:

$A\; =\; \backslash left|\backslash vec\; M\backslash right|\backslash theta$

The angular velocity is usually known $\backslash omega$ in radians per second and torque action time $t$.

Then the work done by the moment of force is calculated as:

$A\; =\; \backslash left|\backslash vec\; M\backslash right|\backslash omega\; t$

Moment of force about a point

If there is a material point $O\_F$, to which the force is applied $\backslash vec\; F$, then the moment of force relative to the point $O$ equal to the vector product of the radius vector $\backslash vec\; r$, connecting the points $O$ And $O\_F$, to the force vector $\backslash vec\; F$:

$\backslash vec(M\_O)\; =\; \backslash left[\backslash vec\; r\; \backslash times\; \backslash vec\; F\backslash right]$.

Moment of force about the axis

The moment of a force relative to an axis is equal to the algebraic moment of the projection of this force onto a plane perpendicular to this axis relative to the point of intersection of the axis with the plane, that is $M\_z(F)\; =\; M\_o(F")\; =\; F"h"$.

Units

The moment of force is measured in newton meters. 1 Nm is the moment produced by a force of 1 N on a lever 1 m long, applied to the end of the lever and directed perpendicular to it.

Torque measurement

Today, the measurement of moment of force is carried out using strain gauges, optical and inductive load cells.

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Excerpt characterizing the Moment of Power

But although by the end of the battle people felt the full horror of their action, although they would have been glad to stop, some incomprehensible, mysterious force still continued to guide them, and, sweating, covered in gunpowder and blood, left one by three, the artillerymen, although and stumbling and gasping from fatigue, they brought charges, loaded, aimed, applied wicks; and the cannonballs flew just as quickly and cruelly from both sides and flattened human body, and that terrible thing continued to happen, which is done not by the will of people, but by the will of the one who leads people and worlds.
Anyone who looked at the upset behinds of the Russian army would say that the French only have to make one more small effort, and the Russian army will disappear; and anyone who looked at the backs of the French would say that the Russians only have to make one more small effort, and the French will perish. But neither the French nor the Russians made this effort, and the flames of the battle slowly burned out.
The Russians did not make this effort because they were not the ones who attacked the French. At the beginning of the battle, they only stood on the road to Moscow, blocking it, and in the same way they continued to stand at the end of the battle, as they stood at the beginning of it. But even if the goal of the Russians was to shoot down the French, they could not make this last effort, because all the Russian troops were defeated, there was not a single part of the troops that was not injured in the battle, and the Russians, remaining in their places , lost half of their army.
The French, with the memory of all the previous victories of fifteen years, with the confidence of Napoleon's invincibility, with the consciousness that they had captured part of the battlefield, that they had lost only one-quarter of their men and that they still had twenty thousand intact guards, it was easy to make this effort. The French, who attacked the Russian army in order to knock it out of position, had to make this effort, because as long as the Russians, just like before the battle, blocked the road to Moscow, the French goal was not achieved and all their efforts and the losses were wasted. But the French did not make this effort. Some historians say that Napoleon should have given his old guard intact in order for the battle to be won. Talking about what would have happened if Napoleon had given his guard is the same as talking about what would have happened if spring had turned into autumn. This couldn't happen. Napoleon did not give his guards, because he did not want it, but this could not be done. All the generals, officers, and soldiers of the French army knew that this could not be done, because the fallen spirit of the army did not allow it.
Napoleon was not the only one who experienced that dream-like feeling that the terrible swing of his arm was falling powerlessly, but all the generals, all the soldiers of the French army who participated and did not participate, after all the experiences of previous battles (where, after ten times less effort, the enemy fled), experienced the same feeling of horror before that enemy who, having lost half the army, stood just as menacingly at the end as at the beginning of the battle. The moral strength of the French attacking army was exhausted. Not the victory that is determined by the pieces of material picked up on sticks called banners, and by the space on which the troops stood and are standing, but a moral victory, one that convinces the enemy of the moral superiority of his enemy and of his own powerlessness, was won by the Russians under Borodin. The French invasion, like an enraged beast that received a mortal wound in its run, felt its death; but it could not stop, just as the twice weaker Russian army could not help but deviate. After this push, the French army could still reach Moscow; but there, without new efforts on the part of the Russian army, it had to die, bleeding from the fatal wound inflicted at Borodino. The direct consequence of the Battle of Borodino was the causeless flight of Napoleon from Moscow, the return along the old Smolensk road, the death of the five hundred thousandth invasion and the death of Napoleonic France, which for the first time at Borodino was laid down by the hand of the strongest enemy in spirit.

Absolute continuity of movement is incomprehensible to the human mind. The laws of any movement become clear to a person only when he examines arbitrarily taken units of this movement. But at the same time, most of human errors stem from this arbitrary division of continuous movement into discontinuous units.
The so-called sophism of the ancients is known, which consists in the fact that Achilles will never catch up with the tortoise in front, despite the fact that Achilles walks ten times faster than the tortoise: as soon as Achilles passes the space separating him from the tortoise, the tortoise will pass ahead of him one tenth of this space; Achilles will walk this tenth, the tortoise will walk one hundredth, etc. ad infinitum. This task seemed insoluble to the ancients. The meaninglessness of the decision (that Achilles would never catch up with the tortoise) stemmed from the fact that discontinuous units of movement were arbitrarily allowed, while the movement of both Achilles and the tortoise was continuous.
By taking smaller and smaller units of movement, we only get closer to the solution of the problem, but never achieve it. Only by admitting an infinitesimal value and an ascending progression from it to one tenth and taking the sum of this geometric progression, we reach a solution to the issue. A new branch of mathematics, having achieved the art of dealing with infinitesimal quantities, and in other more complex questions of motion, now provides answers to questions that seemed insoluble.
This new, unknown to the ancients, branch of mathematics, when considering issues of motion, admits infinitesimal quantities, that is, those at which the main condition of motion is restored (absolute continuity), thereby correcting that inevitable mistake that the human mind cannot help but make when considering instead of continuous movement, individual units of movement.
In the search for the laws of historical movement, exactly the same thing happens.
The movement of humanity, resulting from countless human tyranny, occurs continuously.
Comprehension of the laws of this movement is the goal of history. But in order to comprehend the laws of continuous movement of the sum of all the arbitrariness of people, the human mind allows for arbitrary, discontinuous units. The first technique of history is to take arbitrary series continuous events, consider it separately from others, whereas there is not and cannot be the beginning of any event, but always one event continuously follows from another. The second technique is to consider the action of one person, a king, a commander, as the sum of the arbitrariness of people, while the sum of human arbitrariness is never expressed in the activity of one historical person.
Historical science, in its movement, constantly accepts smaller and smaller units for consideration and in this way strives to get closer to the truth. But no matter how small the units that history accepts, we feel that the assumption of a unit separated from another, the assumption of the beginning of some phenomenon and the assumption that the arbitrariness of all people is expressed in the actions of one historical person are false in themselves.
Every conclusion of history, without the slightest effort on the part of criticism, disintegrates like dust, leaving nothing behind, only due to the fact that criticism selects a larger or smaller discontinuous unit as the object of observation; to which she always has the right, since the historical unit taken is always arbitrary.
Only by allowing an infinitely small unit for observation - the differential of history, that is, the homogeneous drives of people, and having achieved the art of integrating (taking the sums of these infinitesimals), can we hope to comprehend the laws of history.
First fifteen years XIX century in Europe represent an extraordinary movement of millions of people. People leave their usual occupations, rush from one side of Europe to the other, rob, kill one another, triumph and despair, and the whole course of life changes for several years and represents an intensified movement, which at first increases, then weakens. What was the reason for this movement or according to what laws did it occur? - asks the human mind.
Historians, answering this question, describe to us the actions and speeches of several dozen people in one of the buildings in the city of Paris, calling these actions and speeches the word revolution; then they give a detailed biography of Napoleon and some people sympathetic and hostile to him, talk about the influence of some of these people on others and say: this is why this movement occurred, and these are its laws.
But the human mind not only refuses to believe in this explanation, but directly says that the method of explanation is not correct, because with this explanation the weakest phenomenon is taken as the cause of the strongest. The sum of human arbitrariness made both the revolution and Napoleon, and only the sum of these arbitrarinesses tolerated them and destroyed them.

The best definition of torque is the tendency of a force to rotate an object about an axis, fulcrum, or pivot point. Torque can be calculated using force and moment arm (the perpendicular distance from the axis to the line of action of the force), or using moment of inertia and angular acceleration.

Steps

Using force and moment leverage

1. Determine the forces acting on the body and the corresponding moments. If the force is not perpendicular to the moment arm in question (i.e. it acts at an angle), then you may need to find its components using trigonometric functions, such as sine or cosine.

   • The force component considered will depend on the perpendicular force equivalent.
   • Imagine a horizontal rod on which a force of 10 N must be applied at an angle of 30° above the horizontal plane to rotate it about its center.
   • Since you need to use a force that is not perpendicular to the moment arm, you need a vertical component of the force to rotate the rod.
   • Therefore, one must consider the y-component, or use F = 10sin30° N.

2. Use the moment equation, τ = Fr, and simply replace the variables with given or received data.

   • A simple example: Imagine a child weighing 30 kg sitting on one end of a swing board. The length of one side of the swing is 1.5 m.
   • Since the swing's rotation axis is at the center, you don't need to multiply the length.
   • You need to determine the force exerted by the child using mass and acceleration.
   • Since mass is given, you need to multiply it by the acceleration due to gravity, g, equal to 9.81 m/s 2 . Hence:
   • You now have all the necessary data to use the moment equation:

3. Use signs (plus or minus) to show the direction of the moment. If the force rotates the body clockwise, then the moment is negative. If the force rotates the body counterclockwise, then the moment is positive.

   • In the case of several applied forces, simply add up all the moments in the body.
   • Since each force tends to cause different directions of rotation, it is important to use the rotation sign to keep track of the direction of each force.
   • For example, two forces were applied to the rim of a wheel having a diameter of 0.050 m, F 1 = 10.0 N, directed clockwise, and F 2 = 9.0 N, directed counterclockwise.
   • Because the given body– a circle, the fixed axis is its center. You need to divide the diameter and get the radius. The size of the radius will serve as a moment arm. Therefore, the radius is 0.025 m.
   • For clarity, we can solve separate equations for each of the moments arising from the corresponding force.
   • For force 1, the action is directed clockwise, therefore, the moment it creates is negative:
   • For force 2, the action is directed counterclockwise, therefore, the moment it creates is positive:
   • Now we can add up all the moments to get the resulting torque:

   Using moment of inertia and angular acceleration

   1. To start solving the problem, understand how the moment of inertia of a body works. The moment of inertia of a body is the resistance of a body to rotational motion. The moment of inertia depends both on the mass and on the nature of its distribution.

      • To understand this clearly, imagine two cylinders of the same diameter but different masses.
      • Imagine that you need to rotate both cylinders around their central axis.
      • Obviously, a cylinder with more mass will be more difficult to turn than another cylinder because it is “heavier.”
      • Now imagine two cylinders of different diameters, but the same mass. To look cylindrical and have different masses, but at the same time have different diameters, shape, or mass distribution of both cylinders must be different.
      • A cylinder with a larger diameter will look like a flat, rounded plate, while a smaller cylinder will look like a solid tube of fabric.
      • A cylinder with a larger diameter will be more difficult to rotate because you need to apply more force to overcome the longer torque arm.

   2. Select the equation you will use to calculate the moment of inertia. There are several equations that can be used to do this.

      • The first equation is the simplest: the summation of the masses and moment arms of all particles.
      • This equation is used for material points, or particles. An ideal particle is a body that has mass but does not occupy space.
      • In other words, the only significant characteristic of this body is mass; you don't need to know its size, shape or structure.
      • The idea of ​​a material particle is widely used in physics to simplify calculations and use ideal and theoretical schemes.
      • Now imagine an object like a hollow cylinder or a solid uniform sphere. These objects have a clear and defined shape, size and structure.
      • Therefore, you cannot consider them as a material point.
      • Fortunately, you can use formulas that apply to some common objects:

   3. Find the moment of inertia. To start calculating torque, you need to find the moment of inertia. Use the following example as a guide:

      • Two small “weights” with masses of 5.0 kg and 7.0 kg are mounted at a distance of 4.0 m from each other on a light rod (the mass of which can be neglected). The axis of rotation is in the middle of the rod. The rod spins from rest to an angular velocity of 30.0 rad/s in 3.00 s. Calculate the torque produced.
      • Since the axis of rotation is in the middle of the rod, the moment arm of both loads is equal to half its length, i.e. 2.0 m.
      • Since the shape, size and structure of the “loads” are not specified, we can assume that the loads are material particles.
      • The moment of inertia can be calculated as follows:

   4. Find the angular acceleration, α. To calculate angular acceleration, you can use the formula α= at/r.

      • The first formula, α= at/r, can be used when the tangential acceleration and radius are given.
      • Tangential acceleration is acceleration directed tangentially to the direction of motion.
      • Imagine an object moving along a curved path. Tangential acceleration is simply its linear acceleration at any point along the entire path.
      • In the case of the second formula, it is easiest to illustrate it by connecting it with concepts from kinematics: displacement, linear velocity and linear acceleration.
      • Displacement is the distance traveled by an object (SI unit is meters, m); linear velocity is an indicator of the change in displacement per unit of time (SI unit - m/s); linear acceleration is an indicator of the change in linear speed per unit of time (SI unit - m/s 2).
      • Now let's look at the analogs of these quantities in rotational motion: angular displacement, θ - the angle of rotation of a certain point or segment (SI unit - rad); angular velocity, ω – change in angular displacement per unit time (SI unit – rad/s); and angular acceleration, α – change in angular velocity per unit time (SI unit – rad/s 2).
      • Returning to our example, we were given data for angular momentum and time. Since the rotation started from rest, the initial angular velocity is 0. We can use the equation to find:
   5. If you find it difficult to imagine how rotation occurs, then take a pen and try to recreate the task. For more accurate reproduction, do not forget to copy the position of the rotation axis and the direction of the applied force.