In the previous section, we saw that, motion of a rigid body can be: [Translation + 'something else']. In this section, we will see what this 'something else' is.
1. To study about this situation, we first need to constrain the rigid body
• We have to constrain it in such a way that, it cannot have translational motion
2. The most effective way to achieve this, is to fix it along a straight line
• The animation in fig.7.5 below shows an example:
• The small yellow rigid block is attached to the larger red rigid block
3. The attachment is along an edge. This edge is highlighted by a blue line
• This blue line is the 'straight line' along which the yellow block is fixed
• Because of this 'fixity', the yellow block cannot have translational motion
■ When a rigid body is fixed along a line, only one type of motion is possible for that body:
The rotational motion
(If in the above fig.7.5, the large red block was absent, the yellow block can complete a full 360o rotation)
■ The line along which the rigid body is fixed is called it's axis of rotation
• Note that, the axis of rotation need not pass through the geometric center of the body
Example 1: The ceiling fan
An animation of a 'rotating ceiling fan' is shown in fig.7.6 below
• The axis of rotation passes through the center of the ceiling fan. it is shown in fig.7.7 below:
• The axis is indicated by the blue straight line
• The blue curved arrow indicates the direction of rotation
• The fan is fixed along the axis
■ If during rotation, the 'mechanism which fixes the axis' fails, the fan will move with both rotation and translation, causing serious accidents
• So we see that, fixity along the axis is important to ensure pure rotation
Example 2: The potter's wheel
An animation of a rotating potter's wheel is shown in fig.7.8 below:
• The axis of rotation passes through the center of the wheel. It is shown in fig.7.9 below:
• The axis is indicated by the blue straight line
• The curved arrow indicates the direction of rotation
■ If the 'mechanism which fixes the axis' fails, the wheel will move with both rotation and translation
• So we see that, fixity along the axis is important to ensure pure rotation
• The axis of rotation is shown in blue color
■ For convenience of mathematical calculations, we will consider this axis to be the 'z-axis' of the 'frame of reference'
♦ The x-axis is shown in red color
♦ The y-axis is shown in green color
2. We are inclined to think that: Every particle in the body will be in rotation
• But that is not entirely true. Let us see the reason:
• We will consider 3 particles inside the body
♦ They are indicated by small spheres
♦ The red, yellow and green spheres
3. If we trace the path of the red sphere, that path will be a circle
• It is the red circle that we see in the animation
• What is the radius of this circle?
• Obviously, radius = distance of the red sphere from the axis
• We will denote it as rR
4. If we trace the path of the yellow sphere, it will be a circle
• It is the yellow circle that we see in the animation
• What is the radius of this circle?
• Obviously, radius = distance of the yellow sphere from the axis
• We will denote it as rY
5. If we try to trace the path of the green sphere, we will not succeed
• This is because there is no such path
• In other words, there is no green circle. That is., rG = 0
• 'No path' means that, there is no movement. That is., the green sphere is stationary
■ Like the green sphere, all the particles which lie on the axis are stationary. This is the answer to the doubt mentioned in (2) above
6. So we have two circles: The red circle and the yellow circle
• We know that, any circle will be having it's own plane
• The plane in which the red circle lies, is shown in red color in fig.7.11(a) below
• The plane in which the yellow circle lies, is shown blue color in fig.b.
• The plane of the red circle is perpendicular to the axis of rotation
♦ This is indicated by the white set-square in fig.a
• The plane of the yellow circle is also perpendicular to the axis
♦ This is indicated by the white set-square in fig.b
■ If a rigid body is in rotation about an axis, then:
• Every particle of the body, which lies on the axis will be stationary
• Each of the other particles will be rotating in it's own circular path
♦ The center of that circular path lies on the axis
♦ Radius of that circular path = Distance of the particle from the axis
• That circular path lies on a plane
♦ This plane is perpendicular to the axis
1. To study about this situation, we first need to constrain the rigid body
• We have to constrain it in such a way that, it cannot have translational motion
2. The most effective way to achieve this, is to fix it along a straight line
• The animation in fig.7.5 below shows an example:
 |
| Fig.7.5 |
3. The attachment is along an edge. This edge is highlighted by a blue line
• This blue line is the 'straight line' along which the yellow block is fixed
• Because of this 'fixity', the yellow block cannot have translational motion
■ When a rigid body is fixed along a line, only one type of motion is possible for that body:
The rotational motion
(If in the above fig.7.5, the large red block was absent, the yellow block can complete a full 360o rotation)
■ The line along which the rigid body is fixed is called it's axis of rotation
• Note that, the axis of rotation need not pass through the geometric center of the body
In our day to day life, we come across many examples where rigid bodies rotate about an axis. let us see some of them:
An animation of a 'rotating ceiling fan' is shown in fig.7.6 below
 |
| Fig.7.6 |
 |
| Fig.7.7 |
• The blue curved arrow indicates the direction of rotation
• The fan is fixed along the axis
■ If during rotation, the 'mechanism which fixes the axis' fails, the fan will move with both rotation and translation, causing serious accidents
• So we see that, fixity along the axis is important to ensure pure rotation
Example 2: The potter's wheel
An animation of a rotating potter's wheel is shown in fig.7.8 below:
 |
| Fig.7.8 |
 |
| Fig.7.9 |
• The curved arrow indicates the direction of rotation
■ If the 'mechanism which fixes the axis' fails, the wheel will move with both rotation and translation
• So we see that, fixity along the axis is important to ensure pure rotation
Characteristics of rotational motion
1. The animation in fig.7.10 below, shows a rotating body |
| Fig.7.10 |
■ For convenience of mathematical calculations, we will consider this axis to be the 'z-axis' of the 'frame of reference'
♦ The x-axis is shown in red color
♦ The y-axis is shown in green color
2. We are inclined to think that: Every particle in the body will be in rotation
• But that is not entirely true. Let us see the reason:
• We will consider 3 particles inside the body
♦ They are indicated by small spheres
♦ The red, yellow and green spheres
3. If we trace the path of the red sphere, that path will be a circle
• It is the red circle that we see in the animation
• What is the radius of this circle?
• Obviously, radius = distance of the red sphere from the axis
• We will denote it as rR
4. If we trace the path of the yellow sphere, it will be a circle
• It is the yellow circle that we see in the animation
• What is the radius of this circle?
• Obviously, radius = distance of the yellow sphere from the axis
• We will denote it as rY
5. If we try to trace the path of the green sphere, we will not succeed
• This is because there is no such path
• In other words, there is no green circle. That is., rG = 0
• 'No path' means that, there is no movement. That is., the green sphere is stationary
■ Like the green sphere, all the particles which lie on the axis are stationary. This is the answer to the doubt mentioned in (2) above
6. So we have two circles: The red circle and the yellow circle
• We know that, any circle will be having it's own plane
• The plane in which the red circle lies, is shown in red color in fig.7.11(a) below
 |
| Fig.7.11 |
• The plane of the red circle is perpendicular to the axis of rotation
♦ This is indicated by the white set-square in fig.a
• The plane of the yellow circle is also perpendicular to the axis
♦ This is indicated by the white set-square in fig.b
Let us write a summary of the 'characteristics of rotational motion':
• Every particle of the body, which lies on the axis will be stationary
• Each of the other particles will be rotating in it's own circular path
♦ The center of that circular path lies on the axis
♦ Radius of that circular path = Distance of the particle from the axis
• That circular path lies on a plane
♦ This plane is perpendicular to the axis
Also it is worth mentioning that, in the animation in fig.7.10 above, the red sphere moves with a greater speed than the yellow sphere
This is because, rR > rY
We saw the detailed explanation in a previous section.
This is because, rR > rY
We saw the detailed explanation in a previous section.
In the next section, we will see another type of rotational motion
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