# Rolling motion, formula with examples to know.

During our day-to-day activities, we often encounter the rolling motion. Rolling motion is a type of motion that includes the combined motion of rotational and translational motion of an object with respect to a surface in ideal conditions, where no sliding happens

## Rolling motion,

During our day-to-day activities, we often encounter the rolling motion. Rolling motion is a type of motion that includes the combined motion of **rotational **and translational motion of an object with respect to a surface in ideal conditions, where no sliding happens.

An object like a disc, wheel, ball or any circular body that is rolling over a stationary ground surface undergoes rolling motion and it is an important point to note that at every instant, there is a single point of contact without slipping.

Why do such things happen? **Because at any particular instant of time, the part of the disc in contact with the surface is at rest with respect to the surface. Where the point of contact with the ground has zero velocity, which matches the ground velocity and is not slipping.**

For an object, the motion of the center of mass is the motion of that object. The center of mass is a position defined relative to an object or system of objects. During the rolling motion of an object, the surface in contact gets deformed a little bit temporarily.

Due to this deformation, a finite area of both the object and surface comes together. Then what is the effect of this phenomenon on the object and the surface? The effect is that the components of the contact force parallel to the surface oppose** motion resulting in friction**.

Let us discuss how the rolling motion works. Suppose v_{cm} is the velocity of the **center of mass** and is the translational velocity of the disc-like object. As the center of the rolling disc is at its geometric center C, v_{cm }is the velocity of C **(Image **A), which is parallel to the level/rolling surface.

The rotational motion of the object happens about its axis of symmetry, which passes through C. Thus the velocity of any point of the object 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}** and due to rotation, the magnitude of v

_{r}is rω i.e v

_{r}=rω, where

- ω is the angular velocity of the rotation of the disc-like object about the axis.
- r is the distance of the point from the axis (i.e from C)

The velocity v_{r} is perpendicular to the radius vector of the given point with respect to C. At point P_{0} of image A, v_{r} is exactly opposite to v_{cm} due to rotation, where the magnitude of

v_{r} is Rω. where

- R is the radius of the disc-like object.

Hence the condition for the disc-like object to roll without slipping is

**v _{cm}=Rω**.

At P_{1} there are two velocities exist, i.e v_{cm} and v_{r}. Now at P_{1} the top of the object,** the magnitude of velocity is,**

v_{1}=v_{cm}+v_{r}=v_{cm}+Rω=Rω+Rω=2Rω or v_{cm}+v_{cm}=2v_{cm} or

v_{1}=2Rω=2v_{cm}

Such a condition is applicable to all rolling bodies.

**The kinetic energy of the body in a rolling motion**

The kinetic energy of a rolling body can be separated into the kinetic energy of translational motion and the kinetic energy of rotational motion. The kinetic energy of the **translational motion is **given by the following formula.

#### Formula

**K _{T}=mv^{2}_{cm}/2,** where

- m is the mass of the rolling body.
- v
_{cm}is the center of the mass velocity.

Further, the motion of the rolling body about the center of mass in rotation, K_{R }represents the kinetic energy of the rotation of the body;

**K _{R}=Iω^{2}/2**, where

*I*represent the moment of inertia about the axis of symmetry of the rolling body.- ω is the angular velocity.

Now the kinetic energy of the pure rolling body is given by the formula;

#### The formula of the kinetic energy of the pure rolling body;

**K=1/2Iω ^{2}+1/2 mv^{2}_{cm}** where

- m is the mass of the rolling body.
- v
_{cm}is rotation motion. - I represent the moment of inertia.
- ω is the angular velocity of the rolling body.

Then the kinetic energy in terms of the **radius of the gyration **of the body is given by the equation;

**K=1/2 mv ^{2}_{cm}(1+k^{2}/R**

^{2}), where

- k is the corresponding radius of gyration.

## **Rolling motion on an inclined plane**

Let us see image B, a rigid body with radius R is rolling down an inclined plane at an angle θ with the surface. When the said body is placed on an inclined plane tries to slip down, which leads to static friction force that acts upward. The torque is produced due to friction, which rotates the body.

Now consider the linear acceleration of the center of mass as a and angular acceleration as α and the radius of gyration is k. From the above, figures the linear motion of the object is** mg sin θ-f=ma—–1**

For rotational motion net torque is fR is given as Iα, where I represent the moment of inertia about the axis. The equation is;

fR= Iα

**fR=(mk ^{2}) α———2**

As there is no slipping, it is pure rolling and at the point of contact, the body is at rest. Now the condition for pure rolling is **v=Rω.**

Then the equation for linear acceleration** a=Rα ——— 3**

After solving the above three equations,1,2 and 3 for α and f, we get the following **formula,**

#### Formula;

** α =gsin θ/1+k ^{2}/R^{2}**

**f =mg sin θ/1+ k ^{2}/R^{2}**

We may also derive the condition for pure rolling, to avoid slipping,

**f ≤ µ _{s}N**

**g sin θ/1+k ^{2}/R^{2} ≤ µ_{s} mg sin θ**

**µ _{s} ≥ tan θ/1+k^{2}/R^{2} **

Due to the above condition for **µs, **the body rolls without slipping.

**Examples of rolling motion**;

The examples of rolling motion in our day-to-day life are observed frequently. Some of the examples are;

**Paint Roller**– A paint roller is a simple example of rolling motion. A paint roller is dipped into a paint bucket and then rolled onto the wall forth and back to paint the walls.**Ball**– The ball being round in shape experiences rolling motion. When force is applied for its displacement from the original position, it rotates at an axis perpendicular to the direction of force applied.**Ball bearing**– The ball bearing is a rolling element used in rotational devices. Being made of metal ball bearings are encased between two rings that can rotate independently of each other. In most of the mechanisms, the inner ring is attached to a fixed shaft or axle leaving the outer ring to move freely with negligible friction. Almost all motors use this mechanism to date.**Bicycle**– The bicycle is an example of many types of motion like rolling motion, linear motion, rotational motion, etc. When the tires of a bicycle rotate with the rider on top, without slipping it exhibits a rolling motion.**Wheels of a Car**– A car due to acceleration moves in linear motion due to the rolling motion of the tires. It is the rolling motion of the tires that determine the accelerated speed of the car.

**Conclusion;**

Rolling motion is a type of motion that includes the combined motion of the **r**otational** **and translational motion of an object. The body in such motion experiences both translational and linear velocity. Further during rotation as there is no slipping, it is pure rolling and at the point of contact, the body is at rest. The condition for pure rolling is given by the equation **v=Rω.** Such an equation is also applicable to inclined planes.