Rolling motion is a common phenomenon we encounter in our daily lives. For instance, the wheels of vehicles exhibit this type of motion. It is important to note that only round-shaped objects can perform a rolling motion. There are two main types of rolling motion: Pure Rolling and Impure Rolling. Pure rolling can be further categorized into two types: rolling with skidding and rolling with slipping.
Visualize a scenario where a car is in motion, and suddenly, the brakes are applied, causing the wheels to skid. In this scenario, the wheel experiences more translational motion than rotational motion, resulting in a situation known as rolling with skidding. The point on the wheel in contact with the ground moves forward, resulting in backward friction.
Now, consider a situation where the wheel of a car hits a patch of ice and loses traction, causing the wheels to slip. In this case, the wheels experience more rotational motion than translational motion. The point on the wheel in contact with the ground moves backwards, resulting in forward friction.
Finally, pure rolling occurs when the rolling motion is perfect. In pure rolling, the wheels of a car have an optimal balance of translational and rotational motion. The point of contact of the wheel remains stationary, indicating that no friction is acting on the point of contact.
Condition for Pure Rolling
Pure rolling is defined as the motion of a round object without any slipping or skidding at the point of contact between two bodies. For example, consider a disc rolling without slipping on a flat surface. The point of contact of the disc with the ground remains stationary. We know that rolling is a combination of translational and rotational motion.
The translational motion refers to the motion of the disc's centre of mass. The velocity of the centre of mass, which equals the translational motion of the disc, is denoted by v cm . The geometric centre of the disc is also the centre of mass, meaning that v cm is the velocity of point C, which is parallel to the flat surface.
The rotational motion of the disc occurs along the symmetrical axis passing through the disc's centre. Therefore, along any point on the symmetrical axis, the velocity will have two parts: one due to the translational motion and the other is the linear velocity v r due to the rotational motion. The magnitude of the linear velocity v r is given by rω, where r is the distance of the point from the axis, and ω is the angular velocity. The direction of v r is perpendicular to the radius vector connecting the disc's centre (C) and the selected point.
At the point of contact P 0 , the linear velocity v r, due to rotation, acts in a direction opposite to the translational motion v cm . The magnitude of v r at the point P 0 is Rω, where R is the radius of the disc. The condition for the point of contact to be at rest is v cm = Rω. Thus, for a disc, the condition for rolling without slipping is
v cm = Rω
Here, R is the radius of the disc and ω is the angular velocity. The velocity of point P 1 will be equal to v cm + Rω.
Kinetic Energy of Rolling Motion
The kinetic energy of rolling motion can be broken down into the kinetic energy of rotational motion and the kinetic energy of translational motion.
The kinetic energy of translational motion, denoted by K T , is given by mv cm 2 /2, where m is the mass of the rolling object, and v cm is the translational velocity.
The kinetic energy of rotational motion, denoted by K R , is given by Iω 2 /2, where I is the moment of inertia about the symmetrical axis, and ω is the angular velocity.
Therefore, the total kinetic energy of an object in rolling motion is given by
K.E = mv cm 2 /2 + Iω 2 /2
In the case of pure rolling, v cm = Rω. Substituting this into the above equation yields:
K.E = m(Rω) 2 /2 + Iω 2 /2
K.E = ½ [mR 2 + I]ω 2
The kinetic energy in terms of the radius of gyration is given by K.E = (½)mv 2 [1 + k 2 /r 2 ], where k is the radius of gyration.
Pure Rolling on an Inclined Plane
Let's examine a rigid body of radius R rolling down an inclined plane at an angle θ with the horizontal surface. The body, when placed on an inclined plane, tries to slip down, creating static friction that acts upwards. This friction generates torque that causes the body to rotate. Let "a" be the linear acceleration of the centre of mass, α be the angular acceleration of the body, and "k" be the radius of gyration.
The linear motion parallel to the plane is given by
Mgsinθ – f = ma —–(1)
For rotational motion with the axis passing through the centre of mass, the net torque is given by Iα, where "I" is the moment of inertia about the axis.
fR = (mk 2 )α ——-(2)
For pure rolling, the point of contact of the body remains stationary. The condition for pure rolling is v = Rω, and the linear acceleration a = Rα——(3).
Solving equations (1), (2), and (3) for "a" and "f", we get
What is the difference between pure and impure rolling?
The body starts rolling without slipping in pure rolling, as the body’s point of contact with the ground is always at rest, whereas in impure rolling, the body slips as it rolls.
What is the pure rolling condition?
A condition in which there is no sliding during rolling is known as pure rolling. This can only happen when the rolling object’s contact point with the surface has zero velocity. It happens when the velocity of translation motion cancels out the velocity of rotation motion precisely. This will occur if (V = ωr).
Is friction necessary for pure accelerated rolling on an inclined plane?
Friction is required for pure accelerated rolling on an inclined plane.