In our everyday life, one of the most common motions observed is the rolling motion. The wheels used in vehicles demonstrate rolling motion. Rolling motion happens only for round-shaped objects. Rolling motion can be classified into two categories: Pure Rolling and Impure Rolling. Pure rolling can be further divided into two types: rolling with skidding and rolling with slipping.
Suppose a car is moving, and suddenly a brake is applied, and the wheels of the car start skidding. At this time, the wheel hardly rotates but will have more of a translational motion. This is called rolling with skidding. The point on the wheel that is in contact with the ground moves forward. Therefore, the friction acting is backward.
In the next case, imagine that the wheel of the tyre hits a patch of ice and loses traction, and the wheels start slipping. Here, the wheels have more rotational motion than translational motion. The bottom-most or the point on the wheel in contact with the ground moves backwards. Therefore, the friction acts in the forward direction.
Lastly comes pure rolling; it is when the rolling is perfect. In pure rolling, the wheels of the car will have the right amount of translational motion and rotational motion. Here, the point of contact of the wheel will always be at rest. Hence, there is no friction acting on the point of contact.
Condition for Pure Rolling
Pure rolling means the motion of the round object without any slipping or skidding at the point of contact between the two bodies. For example, let us consider a disc rolling without slipping on a level surface. The point of contact of the disc with the ground is at rest. We know that rolling is a combination of translational motion and rotational motion.
The translational motion is the motion of the centre of mass of the disc. The velocity of the centre of mass, which is equal to the translational motion of the disc, is equal to vcm. The centre of mass of the disc is at the geometric centre of the disc. Therefore, vcm is the velocity of point C. This is parallel to the level surface.
The rotational motion of the disc is along the symmetrical axis passing through the centre of the disc. Therefore, along with 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 vr due to the rotational motion. The magnitude of the linear velocity vr is given by rω. Here, r is the distance of the point from the axis, and ω is the angular velocity. The direction of vr is perpendicular to the radius vector joining the centre of the disc (C) and the point taken.
At the point of contact P0, the linear velocity vr, due to rotation, acts in a direction opposite to the translational motion vcm. The magnitude of vr at the point P0 is Rω. Here, R is the radius of the circular disc. The condition for the point of contact to be at rest is vcm = Rω. Thus, for the circular disc, the condition for rolling without slipping is
| vcm = Rω |
R is the radius of the circular disc
ω is the angular velocity
Point P1 will have a velocity equal to vcm + Rω.
Kinetic Energy of Rolling Motion
The kinetic energy of the rolling motion can be separated into the kinetic energy of a rotational motion and the kinetic energy of a translational motion.
The kinetic energy of translational motion is KT = mvcm2/2
Here, m is the mass of the rolling object
vcm is the translational velocity
The kinetic energy of rotational motion is KR = Iω2/2
Here, I is the moment of inertia about the symmetrical axis
ω is the angular velocity
Therefore, the total kinetic energy of an object in rolling motion is
K.E = mvcm2/2 + Iω2/2
In the case of pure rolling, vcm = Rω. Substituting this in the above equation, we get
K.E = m(Rω)2/2 + Iω2/2
K.E = ½ [mR2 + I]ω2
The kinetic energy in terms of the radius of gyration K.E = (½)mv2 [1 + k2/r2]
Where, k is the radius of gyration.
Pure Rolling on an Inclined Plane
Consider a rigid body of radius R that is rolled down an inclined plane at an angle θ with the level surface. The body, when placed on an inclined plane, tries to slip down, and this, in turn, creates static friction that acts upwards. Torque is created because of the friction that causes the body to rotate. Let the linear acceleration of the centre of mass be “a”, the angular acceleration of the body be α, and the radius of gyration be “k”.
From the figure, the linear motion that is parallel to the plane is given as
Mgsinθ – f = ma —–(1)
For rotational motion with the axis passing through the centre of mass, the net torque is given as Iα. Here, “I” is the moment of inertia about the axis.
fR = Iα
fR = (mk2)α ——-(2)
In pure rolling, the point of contact of the body is at rest.
The condition for pure rolling is v = Rω
The linear acceleration a Rα——(3)
Solving equations (1), (2) and (3) for “a” and “f”, we get
Solved Questions – Pure Rolling
1. The following objects are rolled down an inclined plane: solid cylinder, hollow sphere, hollow cylinder and solid sphere. Which object will reach the bottom first?
a. Solid cylinder
b. Hollow sphere
c. Solid sphere
d. Hollow cylinder
Answer: (c) Solid sphere
Solution:
The acceleration is maximum when the moment of inertia (I) is minimum.
The moment of inertia of a solid sphere is minimum, i.e., (⅖)mR2.
Therefore, acceleration is the maximum for a solid sphere. Hence, the solid sphere reaches the bottom first.
2. If a solid sphere is rolled down an inclined plane and a rectangular block of the mass is slipped down a similar inclined plane simultaneously, then
a. They will reach down at the same time
b. The sphere reaches the bottom first
c. The rectangular block reaches first
d. Depends on the density of the object
Answer: (b) The sphere reaches the bottom first
Solution:
Total kinetic energy is given as
K.E= (½)mv2 [1 + k2/r2]
The ratio of k/r is lesser for a solid sphere
Bodies with a smaller ratio of k/r will have a larger velocity and will reach the bottom first.
3. If a solid sphere and a rectangular block of the same mass are rolled down a similar smooth inclined plane,
a. They will reach down at the same time
b. The sphere reaches the bottom first
c. The rectangular block reaches first
d. Depends on the density of the object
Answer: (a) They will reach down at the same time
Solution:
Since the surface of the inclined plane is smooth, both objects will reach the bottom at the same time.
Pure Rolling Video Lesson
Frequently Asked Questions on Pure Rolling
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).
v is the linear velocity.
ω is the angular velocity.
Is friction necessary for pure accelerated rolling on an inclined plane?
Friction is required for pure accelerated rolling on an inclined plane.
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