User:
Mechanisms
Concept
The rotation leads to a change of the angle variation ($\Delta\theta$) which is associated with the final position the angle ($\theta$). Through the radius of rotation, this change is associated with an arc traversed from the distance traveled in a time ($\Delta s$) to the position ($s$).
Mechanisms
ID:(15385, 0)
Angle
Concept
To define a rotation in three-dimensional space, it is necessary first to specify the axis around which the movement will occur. Once the axis has been defined, the angle of rotation that should be applied to the body around that axis can be indicated. It is important to note that the direction of the axis is defined by the straight line that passes through it and, by convention, is usually represented by a unit vector. Likewise, the angle of rotation is measured in radians and can be positive or negative, depending on the direction of rotation that is desired.
ID:(4382, 0)
Describing a Rotation
Concept
When describing a rotational motion, we cannot work with distance in the same way we do when describing translational motion.
• In this case, we must first determine the position of the axis (vector) of rotation.
• Then, we must determine the distance between the object and the axis of rotation.
• Finally, we must estimate the angle of rotation of the object around the axis.
In a rotational motion, the radius remains constant. Any changes in the radius are not part of the rotation, but rather a translation that the object may perform radially.
ID:(4967, 0)
Axis of rotation
Concept
To describe rotation, it is first necessary to determine the axis around which the body rotates:
ID:(10537, 0)
Body rotation
Concept
In some cases, the body must be rotated first before describing the rotation:
ID:(11405, 0)
Rotation of a rotated body
Concept
Once rotated, it is possible to define the axis and describe it in the same way:
ID:(11406, 0)
Three-dimensional body
Concept
In the case of 3D objects, it is necessary to define the axis of rotation in three dimensions, along with the angle that indicates how it rotates around that axis:
ID:(10299, 0)
Fixed axis distance
Concept
The center of the body is not necessarily located on the y-axis, so it is necessary to introduce a distance from the center to the axis:
ID:(10541, 0)
Need to work with radians
Description
If you observe a circle, its perimeter will be $2\pi r$, where
$\displaystyle\frac{\Delta\theta}{2\pi}$
The arc corresponding to the angle $\Delta\theta$ can be calculated as this fraction of the total circumference of the circle:
ID:(9879, 0)
Radians
Concept
In physics, it is common to use radians instead of degrees to measure angles in rotations. This is because in these types of movements, the objects that orbit cover distances that correspond to arcs of a circle. To determine the velocity of the object, it is necessary to calculate the length of the arc covered, which is easy to do if the radius of the orbit and the angle covered in radians are known. For this reason, angles are generally measured in radians to avoid the need for constant conversion between degrees and radians when performing calculations of this type.
ID:(311, 0)
Model
Concept
Variables
Parameters
Selected parameter
Calculations
Equation
$ \Delta s=r \Delta\theta $
Ds = r * Dtheta
$ \Delta s \equiv s - s_0 $
Ds = s - s_0
$ \Delta\theta = \theta - \theta_0 $
Dtheta = theta - theta_0
ID:(15386, 0)
Angle Difference
Equation
To describe the rotation of an object, we need to determine the angle variation ($\Delta\theta$). This is achieved by subtracting the initial Angle ($\theta_0$) from the angle ($\theta$), which is reached by the object during its rotation:
 $ \Delta\theta = \theta - \theta_0 $ |
ID:(3680, 0)
Distance traveled
Equation
We can calculate the distance traveled in a time ($\Delta s$) from the starting position ($s_0$) and the position ($s$) using the following equation:
| $ \Delta s \equiv s - s_0 $ |
ID:(4352, 0)
Arc traveled
Equation
The position the distance traveled in a time ($\Delta s$) in a circular motion can be calculated from the angle variation ($\Delta\theta$) and the radius ($r$) of the orbit using the following formula:
| $ \Delta s=r \Delta\theta $ |
If an object is at a distance equal to the radius of an axis and performs a rotation at an angle
| $ \Delta\theta = \theta - \theta_0 $ |
,
it will have traveled a length
| $ \Delta s \equiv s - s_0 $ |
.
This arc can be calculated by multiplying the radius by the angle, which is
| $ \Delta s=r \Delta\theta $ |
.
ID:(5302, 0)