What are the conditions for pure rolling?

For pure rolling to occur, the following conditions must be satisfied:
1. No Slipping at the Point of Contact:
• The condition for pure rolling is that the point of contact of the rolling object with the surface must have zero velocity relative to the surface. This means that the velocity of the point on the object in contact with the surface should be equal to the velocity of the surface at that point (which is zero).
• Mathematically, this is expressed as: v=Rωv = R \omega where:
o vv is the linear velocity of the center of mass of the object,
o RR is the radius of the object, and
o ω\omega is the angular velocity of the object.
2. Linear and Angular Velocities are Related:
• The linear velocity of the center of mass (vv) and the angular velocity (ω\omega) of the rolling object are related by the above equation. If these velocities do not satisfy this relation, the object is experiencing slipping, which would mean it's not undergoing pure rolling.
3. Static Friction:
• In pure rolling motion, static friction must act at the point of contact between the rolling object and the surface. Static friction ensures that the object rolls without slipping and provides the necessary torque for the angular motion of the object.
• Unlike kinetic friction, which acts when there is sliding between surfaces, static friction in pure rolling does not oppose the motion but helps maintain the rolling condition.
Summary of Conditions for Pure Rolling:
1. v=Rωv = R \omega, where vv is the linear velocity of the center of mass, RR is the radius, and ω\omega is the angular velocity.
2. There is no slipping at the point of contact with the surface.
3. Static friction is involved at the point of contact.
If these conditions are satisfied, the object is said to be in pure rolling motion.