Rolling Motion Physics
You must have seen the motion of a rolling ball or a wheel many times, but do you know the kind of motions that an object and its particles undergo while in rolling motion? A combination of translational and rotational motions happen during the rolling motion of a rigid object. To define rolling motion, we must understand the forces like angular momentum and torque. This article will give you the definition of rolling motion, and you would also learn rolling motion equations here.
Rolling Objects Physics
When there is a rolling motion without slipping, the object has both rotational and translational movement while the point of contact is instantaneously at rest.
(image will be uploaded soon)
Let us first understand pure translational and pure rotational motions.
Pure Translational Motion
An object in pure translational motion has all its points moving with the same velocity as its center of mass i.e. they all have the same speed and direction or V(r) = Vcenter of mass. In the absence of an external force, the object would move in a straight line.
Pure Rotational Motion
An object in pure rolling motion has all its points moving at right angles to the radius (in a plane that is perpendicular to its rotational axis). The speed of these particles is directly proportional to their distance from the axis of rotation. Here V(r) = r * ω. Here ω is the angular frequency. Since at the axis r is 0 hence particles on the axis of rotation do not move at all whereas points at the outer edge move with the highest speed.
(image will be uploaded soon)
The points on either side of the axis of rotation move in opposite directions.
Vpoint of contact = 0 i.e. the point of contact is at rest.
The velocity of the center of mass is Vcenter of mass = R * ω.
The point farthest from the point of contact move with a velocity of
Vopposite the point of contact = 2 * Vcenter of mass = 2 * R * ω.
Heave and Pitch
A ship on the sea has 6 different kinds of motions called:
Heave
This is a linear motion along the vertical z-axis.
Sway
This is also a linear motion along the transverse Y-axis.
Surge
It is again a linear motion along the longitudinal x-axis.
Roll
This is a rotational motion around a longitudinal axis.
Pitch
This is a rotational motion around the transverse axis.
Yaw
This is a rotational motion around the vertical axis.
Mechanical Energy is Conserved in Rolling Motion
As per the rolling motion definition, a rolling object has rotational kinetic energy and translational kinetic energy. If the system requires it, it might also carry potential energy. If we include the gravitational potential energy also then we get the total mechanical energy of a rolling object as:
Etotal = (½ * m * V2center of mass) + (½ * Icenter of mass * ω2) + (m * g * h).
When there are no nonconservative forces that could take away the energy from the system in the form of heat, an object's total energy in rolling motion without slipping is constant throughout the motion. When the object is slipping them energy is not conserved since there is a heat production due to kinetic friction and air resistance.
Moment of Inertia
Rotational inertia is a property of rotating objects. It is the tendency of an object to remain in rotational motion unless a torque is applied to it.
(image will be uploaded soon)
If a force F is exerted on a point mass m at a distance r from the pivot point, then the point mass obtains an acceleration equal to F/m in the direction of F. Since F is perpendicular to r in the case above, the torque τ = F * r. The rotational inertia is given by the formula m * r2.
Parallel Axis Theorem
If the rotational axis passes through the center of mass, then the moment of inertia is minimal. Moment of inertia increases as the distance of the axis of rotation from the center of mass increases. As per the parallel axis theorem, the moment of inertia about an axis that is parallel to the axis across the object’s center of mass is given by the below formula:
Iparallel axis = Icenter of mass + M * d2
Where d is the distance of the parallel axis of rotation from the center of mass.
(image will be uploaded soon)
Let us understand this with an example: Let there be a uniform rod of length l having mass m, rotating about an axis through its center and perpendicular to the rod. What is the moment of inertia Icenter of mass?
Solution. Moment of inertia of a rod = ⅓ * m * l2
Distance of the end of the rod from its center = l/2
Hence the parallel axis theorem of the rod = ( ⅓ * m * l2) - m * (l/2)2)
= ( ⅓ * m * l2) - ( ¼ * m * l2)
Icenter of mass = 1/12 * m * l2
FAQs on Rolling Motion
1. What is rolling motion in physics?
Rolling motion is a type of motion that combines rotation about an object's center of mass with the translation of the center of mass itself. For an object like a wheel or a ball moving along a surface, it simultaneously spins and moves forward. This combination is fundamental to understanding how wheeled objects move.
2. What is the essential condition for 'pure rolling' or rolling without slipping?
The essential condition for pure rolling motion is that the velocity of the center of mass (v_cm) is equal to the product of the object's radius (R) and its angular velocity (ω). This relationship, expressed as v_cm = Rω, ensures that the point of the object in contact with the surface is instantaneously at rest, preventing any slipping or sliding.
3. How can the point of contact be at rest if the object is moving forward?
This is a key concept in rolling motion. The overall motion is a combination of two velocities: a forward translational velocity (+v_cm) for all parts of the object, and a rotational velocity which varies. At the very bottom point of contact, the rotational tangential velocity is directed backward (-Rω). In pure rolling, v_cm = Rω, so these two velocities perfectly cancel each other out (v_cm - Rω = 0), resulting in a point of zero velocity relative to the surface at that instant.
4. What are some common real-world examples of rolling motion?
Rolling motion is observed in many everyday situations. Some common examples include:
- The wheels of a moving car, bicycle, or train.
- A bowling ball rolling down the lane.
- A cylinder or log rolling down a hill.
- A marble or a sphere rolling across a flat floor.
- The motion of a yo-yo as it rolls up and down its string.
5. How is the total kinetic energy of a rolling object calculated?
Since rolling motion is a combination of two motions, its total kinetic energy is the sum of the kinetic energies of translation and rotation. The formula is: K_total = K_translational + K_rotational = (1/2)mv_cm² + (1/2)I_cmω², where 'm' is the mass, 'v_cm' is the velocity of the center of mass, 'I_cm' is the moment of inertia about the center of mass, and 'ω' is the angular velocity.
6. How does the shape of an object affect its rolling motion down an incline?
The shape, which determines the object's moment of inertia (I), significantly affects its motion. An object with a lower moment of inertia (like a solid sphere, where mass is concentrated near the center) will accelerate faster and reach the bottom of an incline first. An object with a higher moment of inertia (like a hollow ring, where mass is far from the center) will accelerate more slowly because more of the potential energy is converted into rotational kinetic energy, leaving less for translational kinetic energy.
7. What is the difference between rolling friction and sliding friction?
Rolling friction (or rolling resistance) is the resistance that occurs when a round object rolls on a surface. It is primarily caused by the non-elastic deformation of the object and the surface at the point of contact. Sliding friction (or kinetic friction) is the force that opposes the motion when two surfaces slide past each other. Generally, for the same object and surface, rolling friction is significantly weaker than sliding friction, which is why it's easier to roll an object than to slide it.
8. What is the importance of the Parallel Axis Theorem in analysing rolling motion?
The Parallel Axis Theorem is crucial when an object rotates about an axis that is not passing through its center of mass. The theorem, I = I_cm + Md², allows us to calculate the moment of inertia (I) about any parallel axis if we know the moment of inertia about the center of mass (I_cm). In rolling motion, this is useful for analysing complex scenarios or calculating the rotational energy about different points, like the point of contact.
