`●` We have seen that every particle of a rotating body moves in a circle. The linear velocity of the particle is related to the angular velocity. The relation between these two quantities involves a vector product .
`●` Let us go back to Fig. 7.4. As said above, in rotational motion of a rigid body about a fixed axis, every particle of the body moves in a circle,
`●` which lies in a plane perpendicular to the axis and has its centre on the axis. In Fig. 7.16 we redraw Fig. 7.4, showing a typical particle (at a point P) of the rigid body rotating about a fixed axis (taken as the z-axis).
`●` The particle describes a circle with a centre C on the axis. The radius of the circle is r, the perpendicular distance of the point P from the axis. We also show the linear velocity vector v of the particle at P. It is along the tangent at P to the circle.
`\color{blue}➢` Let `P′` be the position of the particle after an interval of time `Δt` (Fig. 7.16). The angle `PCP′` describes the angular displacement `Δθ` of the particle in time `Δt`.
`=>` The average angular velocity of the particle over the interval `Δt` is `Δθ//Δt`. As Δt tends to zero (i.e. takes smaller and smaller values), the ratio `Δθ//Δt` approaches a limit which is the instantaneous angular velocity `dθ//dt` of the particle at the position P.
`=>` We denote the instantaneous angular velocity by `ω` (the Greek letter omega). We know from our study of circular motion that the magnitude of linear velocity `v` of a particle moving in a circle is related to the angular velocity of the particle `ω` by the simple relation `υ =ω r` , where `r` is the radius of the circle.
`●` We observe that at any given instant the relation `v =ω r` applies to all particles of the rigid body. Thus for a particle at a perpendicular distance `r_i` from the fixed axis, the linear velocity at a given instant `v_i` is given by
`color{blue}{ω= d theta//dt" " v_i=ω r_i............................(7.19)}`
`=>`The index i runs from `1` to `n`, where n is the total number of particles of the body.
`●` For particles on the axis, ` r = 0` , and hence `v = ω r = 0`. Thus, particles on the axis are stationary. This verifies that the axis is fixed.
`color{green}☞` Note that we use the same angular velocity `ω` for all the particles. We therefore, refer to `ω` as the angular velocity of the whole body. We have characterised pure translation of a body by all parts of the body having the same velocity at any instant of time.
`●` Similarly, we may characterise pure rotation by all parts of the body having the same angular velocity at any instant of time. Note that this characterisation of the rotation of a rigid body about a fixed axis is just another way of saying as in Sec. 7.1 that each particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has the centre on the axis.
`●` In our discussion so far the angular velocity appears to be a scalar. In fact, it is a vector. We shall not justify this fact, but we shall accept it. For rotation about a fixed axis, the angular velocity vector lies along the axis of rotation, and points out in the direction in which a right handed screw would advance, if the head of the screw is rotated with the body. (See Fig. 7.17a).
The magnitude of this vector is referred as above.
`●` We shall now look at what the vector product `ω xx r` corresponds to. Refer to Fig. 7.17(b) which is a part of Fig. 7.16 reproduced to show the path of the particle `P`. The figure shows the vector `ω` directed along the fixed (`z-`) axis and also the position vector `r = OP` of the particle at `P` of the rigid body with respect to the origin `O`. Note that the origin is chosen to be on the axis of rotation.
Now `omega xx r = omega xx OP= omega xx (OC + CP)`
But `omega xx OC=O` as `omega ` is along `OC`
Hence `omega xx r =omega xx CP`
`●` The vector `ω xx CP` is perpendicular to `ω`, i.e. to `CP,` the z-axis and also to the radius of the circle described by the particle at `P`. It is therefore, along the tangent to the circle at `P`. Also, the magnitude of `ω xx CP` is `ω (CP)` since `ω` and `CP` are perpendicular to each other. We shall denote `CP` by `r_⊥` and not by `r`, as we did earlier.
`●` Thus, `ω xx r` is a vector of magnitude `ωr_⊥` and is along the tangent to the circle described by the particle at `P`. The linear velocity vector `v` at `P` has the same magnitude and direction. Thus,
`color{blue}{v= omega xx r....................(7.20)}`
`●` In fact, the relation, Eq. (7.20), holds good even for rotation of a rigid body with one point fixed, such as the rotation of the top [Fig. 7.6(a)]. In this case `r` represents the position vector of the particle with respect to the fixed point taken as the origin.
`●` We note that for rotation about a fixed axis, the direction of the vector `ω` does not change with time. Its magnitude may, however, change from instant to instant. For the more general rotation, both the magnitude and the direction of `ω` may change from instant to instant.
`●` We have seen that every particle of a rotating body moves in a circle. The linear velocity of the particle is related to the angular velocity. The relation between these two quantities involves a vector product .
`●` Let us go back to Fig. 7.4. As said above, in rotational motion of a rigid body about a fixed axis, every particle of the body moves in a circle,
`●` which lies in a plane perpendicular to the axis and has its centre on the axis. In Fig. 7.16 we redraw Fig. 7.4, showing a typical particle (at a point P) of the rigid body rotating about a fixed axis (taken as the z-axis).
`●` The particle describes a circle with a centre C on the axis. The radius of the circle is r, the perpendicular distance of the point P from the axis. We also show the linear velocity vector v of the particle at P. It is along the tangent at P to the circle.
`\color{blue}➢` Let `P′` be the position of the particle after an interval of time `Δt` (Fig. 7.16). The angle `PCP′` describes the angular displacement `Δθ` of the particle in time `Δt`.
`=>` The average angular velocity of the particle over the interval `Δt` is `Δθ//Δt`. As Δt tends to zero (i.e. takes smaller and smaller values), the ratio `Δθ//Δt` approaches a limit which is the instantaneous angular velocity `dθ//dt` of the particle at the position P.
`=>` We denote the instantaneous angular velocity by `ω` (the Greek letter omega). We know from our study of circular motion that the magnitude of linear velocity `v` of a particle moving in a circle is related to the angular velocity of the particle `ω` by the simple relation `υ =ω r` , where `r` is the radius of the circle.
`●` We observe that at any given instant the relation `v =ω r` applies to all particles of the rigid body. Thus for a particle at a perpendicular distance `r_i` from the fixed axis, the linear velocity at a given instant `v_i` is given by
`color{blue}{ω= d theta//dt" " v_i=ω r_i............................(7.19)}`
`=>`The index i runs from `1` to `n`, where n is the total number of particles of the body.
`●` For particles on the axis, ` r = 0` , and hence `v = ω r = 0`. Thus, particles on the axis are stationary. This verifies that the axis is fixed.
`color{green}☞` Note that we use the same angular velocity `ω` for all the particles. We therefore, refer to `ω` as the angular velocity of the whole body. We have characterised pure translation of a body by all parts of the body having the same velocity at any instant of time.
`●` Similarly, we may characterise pure rotation by all parts of the body having the same angular velocity at any instant of time. Note that this characterisation of the rotation of a rigid body about a fixed axis is just another way of saying as in Sec. 7.1 that each particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has the centre on the axis.
`●` In our discussion so far the angular velocity appears to be a scalar. In fact, it is a vector. We shall not justify this fact, but we shall accept it. For rotation about a fixed axis, the angular velocity vector lies along the axis of rotation, and points out in the direction in which a right handed screw would advance, if the head of the screw is rotated with the body. (See Fig. 7.17a).
The magnitude of this vector is referred as above.
`●` We shall now look at what the vector product `ω xx r` corresponds to. Refer to Fig. 7.17(b) which is a part of Fig. 7.16 reproduced to show the path of the particle `P`. The figure shows the vector `ω` directed along the fixed (`z-`) axis and also the position vector `r = OP` of the particle at `P` of the rigid body with respect to the origin `O`. Note that the origin is chosen to be on the axis of rotation.
Now `omega xx r = omega xx OP= omega xx (OC + CP)`
But `omega xx OC=O` as `omega ` is along `OC`
Hence `omega xx r =omega xx CP`
`●` The vector `ω xx CP` is perpendicular to `ω`, i.e. to `CP,` the z-axis and also to the radius of the circle described by the particle at `P`. It is therefore, along the tangent to the circle at `P`. Also, the magnitude of `ω xx CP` is `ω (CP)` since `ω` and `CP` are perpendicular to each other. We shall denote `CP` by `r_⊥` and not by `r`, as we did earlier.
`●` Thus, `ω xx r` is a vector of magnitude `ωr_⊥` and is along the tangent to the circle described by the particle at `P`. The linear velocity vector `v` at `P` has the same magnitude and direction. Thus,
`color{blue}{v= omega xx r....................(7.20)}`
`●` In fact, the relation, Eq. (7.20), holds good even for rotation of a rigid body with one point fixed, such as the rotation of the top [Fig. 7.6(a)]. In this case `r` represents the position vector of the particle with respect to the fixed point taken as the origin.
`●` We note that for rotation about a fixed axis, the direction of the vector `ω` does not change with time. Its magnitude may, however, change from instant to instant. For the more general rotation, both the magnitude and the direction of `ω` may change from instant to instant.