Rigid body physics is the study of many different types of motion. The main ones are translational movement and rotation along a fixed axis. There are also their combinations: free, flat, curvilinear, uniformly accelerated and other varieties. Each movement has its own characteristics, but, of course, there are similarities between them. Consider what kind of movement is called rotational and give examples of such movement, drawing an analogy with translational movement.
The laws of mechanics in action
At first glance, it seems that the rotational movement, examples of which we observe in everyday activities, violates the laws of mechanics. What can be suspected of this violation and what laws?
For example, the law of inertia. Any body, when unbalanced forces do not act on it, must either be at rest or perform uniform rectilinear motion. But if you give the globe a lateral push, it will begin to rotate. Andit would most likely spin forever if it weren't for friction. Like a great example of rotational motion, the globe is constantly rotating, unnoticed by anyone. It turns out that Newton's first law does not apply in this case? It's not.
What moves: a point or a body
Rotational motion is different from forward motion, but there is much in common between them. It is worth comparing and comparing these types, consider examples of translational and rotational motion. To begin with, one should strictly distinguish between the mechanics of a material body and the mechanics of a material point. Recall the definition of translational motion. This is such a movement of the body, in which each of its points moves in the same way. This means that all points of the physical body at each particular moment of time have the same speed in magnitude and direction and describe the same trajectories. Therefore, the translational motion of the body can be considered as the motion of one point, or rather, the motion of its center of mass. If no other bodies act on such a body (material point), then it is at rest, or moves in a straight line and uniformly.
Comparison of formulas for calculation
Examples of the rotational movement of bodies (globe, wheel) show that the rotation of the body is characterized by an angular velocity. It indicates at what angle it will turn per unit of time. In engineering, angular velocity is often expressed in revolutions per minute. If the angular velocity is constant, then we can say that the body rotates uniformly. Whenthe angular velocity increases uniformly, then the rotation is called uniformly accelerated. The similarity of the laws of translational and rotational motions is very significant. Only the letter designations differ, and the calculation formulas are the same. This is clearly seen in the table.
| Forward movement | Rotational movement | |
|
Speed v Path s Time t Acceleration a |
Angular velocity ω Angular movement φ Time t Angular acceleration ą |
|
| s=vt | φ=ωt | |
|
v=at S=at2 / 2 |
ω=ąt φ=ąt2 / 2 |
All tasks in the kinematics of both translational and rotational motion are similarly solved using these formulas.
Role of adhesion force
Let's consider examples of rotational motion in physics. Let's take the movement of one material point - a heavy metal ball from a ball bearing. Is it possible to make it move in a circle? If you push the ball, it will roll in a straight line. You can drive the ball around the circumference, supporting it all the time. But one has only to remove his hand, and he will continue to move in a straight line. From this follows the conclusion that a point can move in a circle only under the action of a force.
This is the movement of a material point, but in a solid body there is not onepoint, but a set. They are connected to each other, as cohesive forces act on them. It is these forces that hold the points in a circular orbit. In the absence of cohesive force, the material points of a rotating body would fly apart like dirt flying off a spinning wheel.
Linear and angular speeds
These examples of rotational motion allow us to draw another parallel between rotational and translational motion. During translational motion, all points of the body move at a certain point in time with the same linear speed. When a body rotates, all its points move with the same angular velocity. In a rotational movement, examples of which are the spokes of a rotating wheel, the angular velocities of all points of the rotating spoke will be the same, but the linear velocities will be different.
Acceleration doesn't count
Recall that in the uniform motion of a point along a circle, there is always an acceleration. Such acceleration is called centripetal. It shows only a change in the direction of speed, but does not characterize the change in speed modulo. Therefore, we can talk about uniform rotational motion with one angular velocity. In engineering, with uniform rotation of the flywheel or rotor of an electric generator, the angular velocity is considered constant. Only a constant number of revolutions of the generator can provide a constant voltage in the network. And this number of revolutions of the flywheel guarantees a smooth and economical running of the machine. Then the rotational motion, examples of which are given above, is characterized only by the angular velocity, without taking into account centripetal acceleration.
Force and its moment
There is another parallel between translational and rotational motion - dynamic. According to Newton's second law, the acceleration received by a body is defined as the division of the applied force by the mass of the body. During rotation, the change in angular velocity depends on the force. Indeed, when screwing a nut, the decisive role is played by the rotating action of the force, and not where this force is applied: to the nut itself or to the wrench handle. Thus, the indicator of force in the formula for translational motion during rotation of the body corresponds to the indicator of the moment of force. Visually, this can be displayed in the form of a table.
| Forward movement | Rotational movement |
| Power F |
Moment of force M=Fl, where l - shoulder strength |
| Work A=Fs | Job A=Mφ |
| Power N=Fs/t=Fv | Power N=Mφ/t=Mω |
Mass of the body, its shape and moment of inertia
The above table does not compare according to the formula of Newton's second law, as this requires additional explanation. This formula includes an indicator of mass, which characterizes the degree of inertia of the body. When a body rotates, its inertia is not characterized by its mass, but is determined by such a quantity as the moment of inertia. This indicator is directly dependent not so much on body weight as on its shape. That is, it matters how the mass of the body is distributed in space. Bodies of various shapes willhave different values of the moment of inertia.
When a material body rotates around a circle, its moment of inertia will be equal to the product of the mass of the rotating body and the square of the radius of the rotation axis. If the point moves twice as far from the axis of rotation, then the moment of inertia and the stability of rotation will increase four times. That's why flywheels are made big. But it is also impossible to increase the radius of the wheel too much, since in this case the centripetal acceleration of the points of its rim increases. The cohesive force of the molecules that forms this acceleration may become insufficient to keep them on a circular path, and the wheel will collapse.
Final comparison
When drawing a parallel between rotational and translational motion, it should be understood that during rotation, the role of body mass is played by the moment of inertia. Then the dynamic law of rotational motion, corresponding to Newton's second law, will say that the moment of force is equal to the product of the moment of inertia and angular acceleration.
Now you can compare all the formulas of the basic equation of dynamics, momentum and kinetic energy in translational and rotational motion, the calculation examples of which are already known.
| Forward movement | Rotational movement |
|
Basic Equation of Dynamics F=ma |
Basic Equation of Dynamics M=Ią |
|
Impulse p=mv |
Impulse p=Iω |
|
Kinetic energy Ek=mv2 / 2 |
Kinetic energy Ek=Iω2 / 2 |
Progressive and rotational movements have much in common. It is only necessary to understand how physical quantities behave in each of these types. When solving problems, very similar formulas are used, the comparison of which is given above.
