In the previous section, we saw the characteristics of 'rotation about an axis'. In this section, we will see another type of rotation.
We will write the steps:
1. The fig.7.12 below shows an animation of a spinning top
• The top is moving around in a circle
• The axis for this circular motion is the blue vertical line shown in fig.7.13 below
♦ This blue axis passes through the 'point of contact of the top with the ground'
• The direction of rotation is indicated by the blue circular arrow
2. While rotating around the blue axis, the top is spinning around it's own axis
• This axis is the orange line shown in fig.7.13
♦ It passes through the center of the top
• The 'spin about this axis' is indicated by the orange circular arrow
3. We can write:
• The blue axis is the Global axis
• The orange axis is the Local axis
■ The top is spinning about it's local axis and at the same time, moving around the global axis
• This movement of the local axis of the top around the global axis is termed as precession
4. In the the previous section,we saw that all the points on the 'axis of the rotating object' will be stationary
• Let us analyse our present case:
(i) The 'axis of the rotating object' in our present case is the orange axis
(ii) Only 'one point on this axis' is stationary
♦ It is the point at the intersection of the orange and blue axes
♦ That is., the point of contact of the top with the ground
(iii) At any instant, the local axis of the top passes through the point of contact
(iv) The point of contact of the top with the ground should not change
• If that point changes, the steps that we wrote above from (1) to (4) become invalid
• Because, then it would mean that, the top has some translation also. The blue line will not be the global axis any more
We will write the steps:
1. The fig.7.12 below shows an animation of a spinning top
 |
| Fig.7.12 |
• The axis for this circular motion is the blue vertical line shown in fig.7.13 below
♦ This blue axis passes through the 'point of contact of the top with the ground'
 |
| Fig.7.13 |
2. While rotating around the blue axis, the top is spinning around it's own axis
• This axis is the orange line shown in fig.7.13
♦ It passes through the center of the top
• The 'spin about this axis' is indicated by the orange circular arrow
3. We can write:
• The blue axis is the Global axis
• The orange axis is the Local axis
■ The top is spinning about it's local axis and at the same time, moving around the global axis
• This movement of the local axis of the top around the global axis is termed as precession
4. In the the previous section,we saw that all the points on the 'axis of the rotating object' will be stationary
• Let us analyse our present case:
(i) The 'axis of the rotating object' in our present case is the orange axis
(ii) Only 'one point on this axis' is stationary
♦ It is the point at the intersection of the orange and blue axes
♦ That is., the point of contact of the top with the ground
(iii) At any instant, the local axis of the top passes through the point of contact
(iv) The point of contact of the top with the ground should not change
• If that point changes, the steps that we wrote above from (1) to (4) become invalid
• Because, then it would mean that, the top has some translation also. The blue line will not be the global axis any more
Let us see another case belonging to this category:
1. The fig.7.14 below shows an animation of an 'oscillating table fan'
• When we consider the total system, the central portion has an oscillating motion
• The path traced by the central portion, during this oscillating motion, is indeed a 'portion of a circle'
• The axis for this circular motion is the blue vertical line shown in fig.7.15 below:
• The direction of rotation is indicated by the blue circular arrow
♦ This arrow is double-headed
♦ This indicates oscillating or to-and-fro motion
2. While the oscillation is taking place around the blue axis, the fan is spinning around it's own axis
• This axis is the orange line shown in fig.7.15
♦ It passes through the center of the fan
• The 'spin about this axis' is indicated by the orange circular arrow
3. We can write:
• The blue axis is the Global axis
• The orange axis is the Local axis
■ The fan is spinning about it's local axis and at the same time, moving around the global axis
4. In the the previous section,we saw that all the points on the 'axis of the rotating object' will be stationary
• Let us analyse our present case:
(i) The 'axis of the rotating object' in our present case is the orange axis
(ii) Only 'one point on this axis' is stationary
♦ It is the point at the intersection of the orange and blue axes
(iii) At any instant, the local axis of the fan passes through the point of intersection
(iv) The point of intersection of the two axes should not change
• If that point changes, the steps that we wrote above from (1) to (4) become invalid
• Because, then it would mean that, the fan has some 'translation' also. The blue line will not be the global axis any more
1. The fig.7.14 below shows an animation of an 'oscillating table fan'
 |
| Fig.7.14 |
• The path traced by the central portion, during this oscillating motion, is indeed a 'portion of a circle'
• The axis for this circular motion is the blue vertical line shown in fig.7.15 below:
 |
| Fig.7.15 |
♦ This arrow is double-headed
♦ This indicates oscillating or to-and-fro motion
2. While the oscillation is taking place around the blue axis, the fan is spinning around it's own axis
• This axis is the orange line shown in fig.7.15
♦ It passes through the center of the fan
• The 'spin about this axis' is indicated by the orange circular arrow
3. We can write:
• The blue axis is the Global axis
■ The fan is spinning about it's local axis and at the same time, moving around the global axis
4. In the the previous section,we saw that all the points on the 'axis of the rotating object' will be stationary
• Let us analyse our present case:
(i) The 'axis of the rotating object' in our present case is the orange axis
(ii) Only 'one point on this axis' is stationary
♦ It is the point at the intersection of the orange and blue axes
(iii) At any instant, the local axis of the fan passes through the point of intersection
(iv) The point of intersection of the two axes should not change
• If that point changes, the steps that we wrote above from (1) to (4) become invalid
• Because, then it would mean that, the fan has some 'translation' also. The blue line will not be the global axis any more
In the 3D animation in fig.7.16 below, a body is moving along a path
Let us write an analysis:
1. The path is a straight line. It is shown in magenta color
2. The motion of the body has one peculiarity:
■ At any instant, the 'center of mass' of the body lies on the magenta line
• This can be explained using the 7 steps given below:
(i) The body has it's own 3 local axes
(ii) The local z-axis of the body is shown to be straight up (the blue arrow) in the 3D view in fig.7.17 below:
• The local x (red arrow) and y (green arrow) axes are also shown
(iii) The direction of motion is indicated by the yellow arrow
(iv)The 'center of mass' is the white sphere
(v) The 'center of mass' coincides with the origin O of the three axes
(vi) This 'center of mass' always lies on the magenta line
(vii) This is shown in the 2D side view in fig.7.18 below also:
3. In the fig.7.18, the different positions of 'center of mass' O are denoted as OA, OB, OC so on . . .
• P is an arbitrary point on the local z-axis
♦ The different positions of P are denoted as PA, PB, PC so on . . .
• We can write the following 3 facts:
(i) In fig.a, the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}=\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OAPA) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}=\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OBPB) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}=\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OCPC) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of both OA and PA are the same
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of both OB and PB are the same
♦ The local z-axis of the body (line OBPB) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of both OC and PC are the same
♦ The local z-axis of the body (line OCPC) makes an angle of α(C) with the horizontal
• We see that: α(A) = α(B) = α(C)
■ That means: The orientation of the body always remains the same
 |
| Fig.7.16 |
1. The path is a straight line. It is shown in magenta color
2. The motion of the body has one peculiarity:
■ At any instant, the 'center of mass' of the body lies on the magenta line
• This can be explained using the 7 steps given below:
(i) The body has it's own 3 local axes
(ii) The local z-axis of the body is shown to be straight up (the blue arrow) in the 3D view in fig.7.17 below:
 |
| Fig.7.17 |
(iii) The direction of motion is indicated by the yellow arrow
(iv)The 'center of mass' is the white sphere
(v) The 'center of mass' coincides with the origin O of the three axes
(vi) This 'center of mass' always lies on the magenta line
(vii) This is shown in the 2D side view in fig.7.18 below also:
 |
| Fig.7.18 |
• P is an arbitrary point on the local z-axis
♦ The different positions of P are denoted as PA, PB, PC so on . . .
• We can write the following 3 facts:
(i) In fig.a, the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}=\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OAPA) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}=\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OBPB) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}=\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OCPC) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of both OA and PA are the same
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of both OB and PB are the same
♦ The local z-axis of the body (line OBPB) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of both OC and PC are the same
♦ The local z-axis of the body (line OCPC) makes an angle of α(C) with the horizontal
• We see that: α(A) = α(B) = α(C)
■ That means: The orientation of the body always remains the same
Let us check whether this is true when the path is curved:
Consider fig.7.19 below:
• Just as in the case of the straight line path, here also we can write the 3 facts
• Steps are the same. However, we will write them again
• The reader is advised to read all 3 steps carefully and become convinced that, they are all valid with respect to fig.7.19
(i) In fig.7.19(a), the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}=\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OAPA) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}=\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OBPB) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}=\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OCPC) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of both OA and PA are the same
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of both OB and PB are the same
♦ The local z-axis of the body (line OBPB) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of both OC and PC are the same
♦ The local z-axis of the body (line OCPC) makes an angle of α(C) with the horizontal
• We see that: α(A) = α(B) = α(C)
Consider fig.7.19 below:
 |
| Fig.7.19 |
• Steps are the same. However, we will write them again
• The reader is advised to read all 3 steps carefully and become convinced that, they are all valid with respect to fig.7.19
(i) In fig.7.19(a), the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}=\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OAPA) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}=\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OBPB) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be parallel to the path
• Since the two velocities are parallel, they have the same direction
• Since the body is rigid, the magnitudes will also be the same
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}=\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OCPC) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of both OA and PA are the same
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of both OB and PB are the same
♦ The local z-axis of the body (line OBPB) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of both OC and PC are the same
♦ The local z-axis of the body (line OCPC) makes an angle of α(C) with the horizontal
• We see that: α(A) = α(B) = α(C)
■ That means: The orientation of the body always remains the same
• The cases in both figs.7.18 and 7.19 that we saw above are pure translation. Now we will see another type:
• In the 3D animation in fig.7.20 below, a body is moving along a path
Let us write an analysis:
1. The path is a straight line. It is shown in magenta color
2. The motion of the body has two peculiarities:
(i) At any instant, the 'center of mass' of the body lies on the magenta line
(ii) The body is spinning about the local y-axis passing through the 'center of mass'
• These can be explained using the 9 steps given below:
(i) The body has it's own 3 local axes
(ii) The local z-axis of the body is shown to be straight up in the 3D view in fig.7.21 below:
• The local x (red arrow) and y (green arrow) axes are also shown
(iii) The direction of motion is indicated by the yellow arrow
(iv) The direction of spin is indicated by the green curved arrow
(v) The body is spinning about the local y-axis
(vi) The 'center of mass' is the white sphere
(vii) The 'center of mass' coincides with the origin O of the three axes
(viii) This 'center of mass' always lies on the magenta line
(ix) This is shown in the 2D side view in fig.7.22 below also:
3. The different positions of O are denoted as OA, OB, OC so on . . .
• P is an arbitrary point in the body. It lies on the local z-axis
♦ The different positions of P are denoted as PA, PB, PC so on . . .
• We can write the following 3 facts:
(i) In fig.a, the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}\neq\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OP) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}\neq\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}\neq\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of OA and PA are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of OB and PB are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of OC and PC are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(C) with the horizontal
• We see that: α(A) ≠ α(B) ≠ α(C)
■ That means: The orientation of the body always changes
Consider fig.7.23 below:
• Just as in the case of the straight line path, here also we can write the 3 facts
• Steps are the same. However, we will write them again
• The reader is advised to read all those steps carefully and become convinced that, they are all valid with respect to fig.7.23
(i) In fig.7.23(a), the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}\neq\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OP) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}\neq\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}\neq\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of OA and PA are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of OB and PB are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of OC and PC are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(C) with the horizontal
• We see that: α(A) ≠ α(B) ≠ α(C)
■ That means: The orientation of the body always changes
• In the 3D animation in fig.7.20 below, a body is moving along a path
 |
| Fig.7.20 |
1. The path is a straight line. It is shown in magenta color
2. The motion of the body has two peculiarities:
(i) At any instant, the 'center of mass' of the body lies on the magenta line
(ii) The body is spinning about the local y-axis passing through the 'center of mass'
• These can be explained using the 9 steps given below:
(i) The body has it's own 3 local axes
(ii) The local z-axis of the body is shown to be straight up in the 3D view in fig.7.21 below:
 |
| Fig.7.21 |
(iii) The direction of motion is indicated by the yellow arrow
(iv) The direction of spin is indicated by the green curved arrow
(v) The body is spinning about the local y-axis
(vi) The 'center of mass' is the white sphere
(vii) The 'center of mass' coincides with the origin O of the three axes
(viii) This 'center of mass' always lies on the magenta line
(ix) This is shown in the 2D side view in fig.7.22 below also:
 |
| Fig.7.22 |
• P is an arbitrary point in the body. It lies on the local z-axis
♦ The different positions of P are denoted as PA, PB, PC so on . . .
• We can write the following 3 facts:
(i) In fig.a, the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}\neq\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OP) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}\neq\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}\neq\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of OA and PA are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of OB and PB are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of OC and PC are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(C) with the horizontal
• We see that: α(A) ≠ α(B) ≠ α(C)
■ That means: The orientation of the body always changes
Let us check whether this is true when the path is curved:
Consider fig.7.23 below:
 |
| Fig.7.23 |
• Steps are the same. However, we will write them again
• The reader is advised to read all those steps carefully and become convinced that, they are all valid with respect to fig.7.23
(i) In fig.7.23(a), the body is at 'Position A'
■ Consider the instant at which the body just passes 'Position A'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(A)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(A)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(A)}}\neq\vec{v}_{\text{O(A)}}}$
■ Now we need a prominent direction to fix up the orientation of the body
• Let us choose the 'horizontal'
• The angle which the local z-axis of the body (line OP) makes with the horizontal is denoted as α(A)
♦ This is shown separately in fig.a
(ii) In fig.b, the body is at 'Position B'
■ Consider the instant at which the body just passes 'Position B'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(B)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(B)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(B)}}\neq\vec{v}_{\text{O(B)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(B)
♦ This is shown separately in fig.b
(iii) In fig.c, the body is at 'Position C'
■ Consider the instant at which the body just passes 'Position C'
• The instantaneous velocity of the 'particle at O' at that instant is $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This velocity will be parallel to the path
♦ Further, the 'particle at O' lies in the path
♦ So we can say: This velocity will be tangential to the path
• The instantaneous velocity of the 'particle at P' at that instant is $\mathbf\small{\vec{v}_{\text{P(C)}}}$
♦ This velocity will be different from $\mathbf\small{\vec{v}_{\text{O(C)}}}$
♦ This is because, the body has both translation and rotation
• So we can write: $\mathbf\small{\vec{v}_{\text{P(C)}}\neq\vec{v}_{\text{O(C)}}}$
■ The angle which local z-axis of the body (line OP) makes with the horizontal is denoted as α(C)
♦ This is shown separately in fig.c
Summary of the 3 facts can be written:
• At position A:
♦ Velocities of OA and PA are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(A) with the horizontal
• At position B:
♦ Velocities of OB and PB are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(B) with the horizontal
• At position C:
♦ Velocities of OC and PC are different
♦ The local z-axis of the body (line OAPA) makes an angle of α(C) with the horizontal
• We see that: α(A) ≠ α(B) ≠ α(C)
■ That means: The orientation of the body always changes
Thus we have two cases:
Case 1:
• This case is explained based on figs.7.16, 7.17, 7.18 and 7.19
• At any instant, all the particles of the body have the same velocity
• The orientation of the body remains the same
• This motion is pure translation
Case 2:
• This case is explained based on figs.7.20, 7.21, 7.22 and 7.23
• At any instant, various particles of the body have different velocities
• The orientation of the body changes continuously
• This motion is [Translation + Something else]
• We see that, the difference between the two cases is:
• In case 2, there is rotational motion about a local axis
• So the 'something else' is: Rotational motion
Case 1:
• This case is explained based on figs.7.16, 7.17, 7.18 and 7.19
• At any instant, all the particles of the body have the same velocity
• The orientation of the body remains the same
• This motion is pure translation
Case 2:
• This case is explained based on figs.7.20, 7.21, 7.22 and 7.23
• At any instant, various particles of the body have different velocities
• The orientation of the body changes continuously
• This motion is [Translation + Something else]
• We see that, the difference between the two cases is:
• In case 2, there is rotational motion about a local axis
• So the 'something else' is: Rotational motion
In rotational motion, we saw two cases:
Case 1:
• Rotational motion in which every point in the axis is fixed
Examples: Ceiling fan, Potter's wheel
Case 2:
• Rotational motion in which only one point in the axis is fixed
Examples: Spinning top, Oscillating fan
• In this chapter, we will be dealing with case 1 only
Case 1:
• Rotational motion in which every point in the axis is fixed
Examples: Ceiling fan, Potter's wheel
Case 2:
• Rotational motion in which only one point in the axis is fixed
Examples: Spinning top, Oscillating fan
• In this chapter, we will be dealing with case 1 only
In the above discussion, we see that,'center of mass' has an important role to play. In the next section, we will see the methods to find it
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