Rotational motion: examples, formulas

Solid state physics studies many different kinds of motion. The main ones are translational motion and rotation along a fixed axis. There are also combinations of them: free, flat, curved, uniformly accelerated and other varieties. Each movement has its own characteristics, but of course there are similarities between them. Consider what kind of motion is called rotational and give examples of such a motion, drawing an analogy with the translational motion.

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 make a uniform rectilinear movement. But if you give the globe a side push, it will begin to spin. And he would most likely spin forever if not for friction. Like a great example of rotational motion - the globe - rotates constantly, not pushed by anyone. It turns out that Newton’s first law does not apply in this case? This is not true.

What moves: point or body

Rotational motion differs from translational, 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 any given moment in time have the same speed and modulus 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 other bodies do not act on such a body (material point), then it is at rest, or moves rectilinearly and evenly.

Comparison of formulas for calculating

Examples of rotational motion of bodies (globe, wheel) show that the rotation of the body is characterized by angular velocity. It indicates at what angle it will turn in a unit of time. In technology, angular velocity is often expressed in revolutions per minute. If the angular velocity is constant, then we can say that the body rotates uniformly. When the angular velocity increases uniformly, the rotation is called uniformly accelerated. The similarity of the laws of translational and rotational movements is very significant. Only letter designations differ, and the calculation formulas are the same. This is clearly visible in the table.

All kinematics problems of both translational and rotational motion are similarly solved by these formulas.

Traction role

Consider examples of rotational motion in physics. 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, then it will roll in a straight line. You can guide the ball around the circumference, supporting it all the time. But you just have to remove your hand, and he will continue to move in a straight line. It follows from this that a point can move in a circle only under the action of force.

This is the movement of a material point, but in a solid, not one point, but many. They are interconnected, since they are affected by adhesion forces. It is these forces that hold the points in a circular orbit. In the absence of adhesion, the material points of the rotating body would scatter as dirt flies off the rotating wheel.

Linear and angular speeds

These examples of rotational motion allow us to draw another parallel between rotational and translational motion. During the translational motion, all points of the body move at a certain moment in time with the same linear speed. When the body rotates, all its points move with the same angular velocity. In rotational motion, examples of which are spokes of a rotating wheel, the angular velocities of all points of the rotating spokes will be the same, and linear ones will be different.

Acceleration doesn't count

Recall that in the uniform motion of a point in a circle there is always acceleration. This 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 technology, 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 such a speed 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 angular velocity, without taking into account centripetal acceleration.

Strength and its moment

There is another parallel between translational and rotational motion - dynamic. As Newton’s second law says, the acceleration received by the 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. After all, when tightening the nut, the decisive role is played by the rotational effect 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. This can be visually displayed in the form of a table.

Body mass, its shape and moment of inertia

The table below does not compare according to the formula of Newton’s second law, since this requires additional explanation. This formula includes a mass indicator that characterizes the degree of inertia of the body. When the body rotates, its inertness is not characterized by its mass, but is determined by such a value as the moment of inertia. This indicator is directly dependent not only on body weight, but on its shape. That is, how the mass of the body is distributed in space matters. Bodies of various shapes will have different values ​​of the moment of inertia.

When a material body rotates around its circumference, the moment of inertia will be equal to the product of the mass of the rotating body by the square of the radius of the axis of rotation. If the point moves twice the distance from the axis of rotation, the moment of inertia index and rotation stability will increase four times. This is why flywheels are made large. But the wheel radius cannot be increased too much, since the centripetal acceleration of the points of its rim increases. The bonding force of the molecules forming 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 moment of inertia plays the role of body mass. 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 during translational and rotational motion, examples of which are already known.

The translational 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.