`●` All wheels used in transportation have rolling motion. For specificness we shall begin with the case of a disc, but the result will apply to any rolling body rolling on a level surface. We shall assume that the disc rolls without slipping.
`●` This means that at any instant of time the bottom of the disc which is in contact with the surface is at rest on the surface.
We have remarked earlier that rolling motion is a combination of rotation and translation. We know that the translational motion of a system of particles is the motion of its centre of mass.
`=>` Let `v_(cm)` be the velocity of the centre of mass and therefore the translational velocity of the disc. Since the centre of mass of the rolling disc is at its geometric centre C (Fig. 7. 37), `v_(cm)` is the velocity of `C`.
`●` It is parallel to the level surface. The rotational motion of the disc is about its symmetry axis, which passes through `C`.
`●` Thus, the velocity of any point of the disc, like `P_0, P_1` or `P_2`, consists of two parts, one is the translational velocity `v_(cm)` and the other is the linear velocity `v_r` on account of rotation. The magnitude of `v_r` is `v_r = rω`, where `ω` is the angular velocity of the rotation of the disc about the axis and `r` is the distance of the point from the axis (i.e. from `C`).
`●` The velocity `v_r` is directed perpendicular to the radius vector of the given point with respect to `C`. In Fig. 7.37, the velocity of the point `P_2 (v_2)` and its components `v_r` and `v_(cm)` are shown; `v_r` here is perpendicular to `CP_2` . It is easy to show that `v_z` is perpendicular to the line `P_OP_2`.
`●` Therefore the line passing through `P_O` and parallel to `ω` is called the instantaneous axis of rotation.
`●` At `P_o`, the linear velocity, `v_r`, due to rotation is directed exactly opposite to the translational velocity `v_(cm)`. Further the magnitude of `v_r` here is `Rω`, where `R` is the radius of the disc. The condition that `P_o` is instantaneously at rest requires `v_(cm) = R_ω`. Thus for the disc the condition for rolling without slipping is ..................... (7.47)
`\color{green} ✍️` Incidentally, this means that the velocity of point `P_1` at the top of the disc `(v_1)` has a magnitude `v_(cm)+ R_ω` or `2 v_(cm)` and is directed parallel to the level surface. The condition (7.47) applies to all rolling bodies.
`●` All wheels used in transportation have rolling motion. For specificness we shall begin with the case of a disc, but the result will apply to any rolling body rolling on a level surface. We shall assume that the disc rolls without slipping.
`●` This means that at any instant of time the bottom of the disc which is in contact with the surface is at rest on the surface.
We have remarked earlier that rolling motion is a combination of rotation and translation. We know that the translational motion of a system of particles is the motion of its centre of mass.
`=>` Let `v_(cm)` be the velocity of the centre of mass and therefore the translational velocity of the disc. Since the centre of mass of the rolling disc is at its geometric centre C (Fig. 7. 37), `v_(cm)` is the velocity of `C`.
`●` It is parallel to the level surface. The rotational motion of the disc is about its symmetry axis, which passes through `C`.
`●` Thus, the velocity of any point of the disc, like `P_0, P_1` or `P_2`, consists of two parts, one is the translational velocity `v_(cm)` and the other is the linear velocity `v_r` on account of rotation. The magnitude of `v_r` is `v_r = rω`, where `ω` is the angular velocity of the rotation of the disc about the axis and `r` is the distance of the point from the axis (i.e. from `C`).
`●` The velocity `v_r` is directed perpendicular to the radius vector of the given point with respect to `C`. In Fig. 7.37, the velocity of the point `P_2 (v_2)` and its components `v_r` and `v_(cm)` are shown; `v_r` here is perpendicular to `CP_2` . It is easy to show that `v_z` is perpendicular to the line `P_OP_2`.
`●` Therefore the line passing through `P_O` and parallel to `ω` is called the instantaneous axis of rotation.
`●` At `P_o`, the linear velocity, `v_r`, due to rotation is directed exactly opposite to the translational velocity `v_(cm)`. Further the magnitude of `v_r` here is `Rω`, where `R` is the radius of the disc. The condition that `P_o` is instantaneously at rest requires `v_(cm) = R_ω`. Thus for the disc the condition for rolling without slipping is ..................... (7.47)
`\color{green} ✍️` Incidentally, this means that the velocity of point `P_1` at the top of the disc `(v_1)` has a magnitude `v_(cm)+ R_ω` or `2 v_(cm)` and is directed parallel to the level surface. The condition (7.47) applies to all rolling bodies.