some children are stand around four side of a rotation table around its axis if all hidren come towards the centre what will be the effects on the motion of table and why
Answers
Explanation:
Understand the relationship between force, mass and acceleration.
Study the turning effect of force.
Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.
If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in [link]. In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.
Force is required to spin the bike wheel. The greater the force, the greater the angular acceleration produced. The more massive the wheel, the smaller the angular acceleration. If you push on a spoke closer to the axle, the angular acceleration will be smaller.
The given figure shows a bike tire being pulled by a hand with a force F backward indicated by a red horizontal arrow that produces an angular acceleration alpha indicated by a curved yellow arrow in counter-clockwise direction.
To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point, as shown in [link]. Because the force is perpendicular to r, an acceleration a=\frac{F}{m} is obtained in the direction of F. We can rearrange this equation such that F=\text{ma} and then look for ways to relate this expression to expressions for rotational quantities. We note that a=\mathrm{r\alpha }, and we substitute this expression into F=\text{ma}, yielding
F=\text{mr}\alpha \text{.}
Recall that torque is the turning effectiveness of a force. In this case, because \mathbf{\text{F}} is perpendicular to r, torque is simply \tau =\mathrm{Fr}. So, if we multiply both sides of the equation above by r, we get torque on the left-hand side. That is,
An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r, causing it to accelerate about the pivot point. The force is kept perpendicular to r.
The given figure shows an object of mass m, kept on a horizontal frictionless table, attached to a pivot point, which is in the center of the table, by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r, which is indicated by a red arrow tangential to the circle, causing the object to move in counterclockwise direcion.
Making Connections: Rotational Motion Dynamics
Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.
Rotational Inertia and Moment of Inertia
Before we can consider the rotation of anything other than a point mass like the one in [link], we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia
I of an object to be the sum of
{\text{mr}}^{2} for all the point masses of which it is composed. That is,
I=\sum {\text{mr}}^{2}. Here
I is analogous to
m in translational motion. Because of the distance
r, the moment of inertia for any object depends on the chosen axis. Actually, calculating
I is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore
{\text{MR}}^{2}, where
M is its total mass and R its radius. (We use M and R for an entire object to distinguish them from m and r for point masses.) In all other cases, we must consult [link] (note that the table is piece of artwork that has shapes as well as formulae) for formulas for I that have been derived from integration over the continuous body. Note that I has units of mass multiplied by distance squared (\text{kg}\cdot {\text{m}}^{2}), as we might expect from its definition.
The general relationship among torque, moment of inertia, and angular acceleration is
where net \tau is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation.