Acceleration and Friction in Pure Rolling Motion on Inclined Plane
Pure rolling on inclined plane is a key concept in JEE Main Physics. It describes the motion of a body, like a cylinder or ring, moving down an incline such that its contact point with the surface does not slip. In this case, translational and rotational motion are perfectly coordinated. Understanding the conditions for pure rolling, acceleration formulae, friction direction, and key numericals helps students solve important exam problems. This page covers each aspect with formulae, diagrams, solved examples, and links to deeper topics, following Vedantu’s JEE methodology.
When a solid object rolls down an inclined surface, like a disc or a ring, there can be pure rolling or slipping. In pure rolling motion on inclined plane, the point of contact with the surface has zero velocity relative to the incline at every instant. Here, the famous condition v = rω holds, where v is the center’s velocity, r is radius, and ω is angular velocity.
Contrast this with slipping—when the v ≠ rω condition is violated and the contact point moves relative to the plane. Many JEE problems focus on identifying which case applies and what it changes about forces, friction, and motion. Mastering this helps with both conceptual and numerical sections, especially for rolling bodies of different shapes.
To deepen your basics, also review rotational motion and definitions in laws of motion. You’ll use these ideas throughout this topic.
Physical Conditions and Formulae for Pure Rolling on Inclined Plane
JEE problems almost always ask for the pure rolling condition, acceleration, and sometimes the forces involved. The primary kinematic condition is v = rω at all times. For a body rolling down a rough incline of angle θ:
| Parameter | Symbol | Formula/Unit |
|---|---|---|
| Pure rolling condition | v = rω | m/s |
| Linear acceleration (for rough incline) | a | a = g sinθ / (1 + k2/r2) |
| Moment of inertia coefficient | k | I = mk2 |
| Friction force (static) | f | f = (mk2g sinθ) / (r2 + k2) |
| Angular acceleration | α | α = a/r |
Here, g is acceleration due to gravity, I is moment of inertia, and k is the radius of gyration. For common bodies:
- Solid sphere: k2 = (2/5) r2
- Hollow sphere: k2 = (2/3) r2
- Cylinder/disc: k2 = (1/2) r2
- Ring: k2 = r2
For a solid sphere, the acceleration becomes a = (5/7) g sinθ. For a ring, it is a = (1/2) g sinθ. These ready-to-use formulas are vital for JEE Main numericals.
For mixed translation and rotation, consult advanced examples in combined translation and rotational motion for deeper insights.
Role and Direction of Friction in Pure Rolling on Inclined Plane
An often-misunderstood aspect in pure rolling motion on inclined plane is friction. In pure rolling, static friction is essential to maintain the no-slip condition, even when it does no work overall. The direction of friction is especially important for JEE conceptual questions.
On a rough incline, friction acts up the plane when rolling is caused by gravity. If an external torque or force tries to increase ω faster than v/r, friction can act down the incline. Friction ensures the v = rω condition holds during pure rolling motion.
- Friction is necessary for pure rolling acceleration to occur.
- Friction does zero work in pure rolling, as point of contact is momentarily at rest.
- For a smooth incline (frictionless), pure rolling is not possible unless ω and v are externally linked.
- Changing incline angle θ affects both friction magnitude and rolling acceleration.
Force analysis in such JEE scenarios often requires free body diagrams. For practice, see static and kinetic friction and block-on-block friction problems to reinforce these ideas.
Numerical Example: Acceleration in Pure Rolling for Ring, Disc, and Sphere
Let’s work through a typical JEE Main problem for pure rolling acceleration formula and friction on an inclined plane. Suppose a solid sphere, solid cylinder, and ring of equal mass and radius start from rest and roll without slipping down an incline of angle θ. Calculate the acceleration for each.
- For a solid sphere: k2 = (2/5) r2
a = g sinθ / (1 + 2/5) = (5/7) g sinθ. - For a solid cylinder: k2 = (1/2) r2
a = g sinθ / (1 + 1/2) = (2/3) g sinθ. - For a ring: k2 = r2
a = g sinθ / (1 + 1) = (1/2) g sinθ.
So, the sphere reaches the bottom first since it has the greatest acceleration, a = (5/7)g sinθ.
For more on energy division in rolling, study kinetic energy of a rotating body and work energy and power.
Common Pitfalls and Best Practices for Pure Rolling on Inclined Plane
Pure rolling does not occur on a frictionless plane unless initial v and ω are set to match v = rω. Many students mistakenly think “frictionless” means pure rolling is easier; it isn’t possible without static friction. Here are classic mistakes and exam tips:
- Always apply v = rω, not v = ω/r.
- Check the direction of friction—often up the incline for gravity-driven cases.
- Remember, friction force exists but work done by friction is zero.
- If given a smooth plane in the question, assume sliding occurs unless told otherwise.
- Acceleration, friction, and energy formulas differ for ring, disc, and sphere; always use the correct k2 value.
Find detailed derivations in moment of inertia and revisit rotational motion revision notes before practicing combined numericals.
Applications of Pure Rolling Motion on Inclined Plane
Pure rolling isn’t just theory—it appears in experiments and real-world JEE problems. Classic examples include:
- Rolling motion of a ring, disc, or sphere in lab setups and mechanics problems.
- Understanding dynamics of ball bearings or wheels on ramps.
- Checking energy conservation using both translational and rotational energies.
- Comparison of time taken to reach the bottom for different shapes.
- Practical limits: sports physics (cricket ball, car tires, skateboards).
For more hands-on experiment examples, try problems involving projectiles on an inclined plane or angle of repose analysis.
Before wrapping up, practice is key. Use relevant JEE exercises in vedantu’s solved physics questions and laws of motion mock tests for timed practice.
This page is written and reviewed by Vedantu’s physics team, drawing on IIT alumni expertise and JEE faculty insights.
FAQs on Pure Rolling on an Inclined Plane: Concepts, Formulas & Application
1. What is pure rolling on an inclined plane?
Pure rolling on an inclined plane occurs when a body moves so that its point of contact with the surface does not slip at any instant. This means both translational and rotational motion are synchronized. Key points include:
- The condition for pure rolling is v = rω, where v is the velocity of the center of mass and ω is the angular velocity.
- The point of contact with the plane is momentarily at rest relative to the surface.
- Pure rolling is common in cases like a wheel, cylinder, ring, or sphere moving down a rough incline.
2. What is the acceleration formula for pure rolling on an inclined plane?
The acceleration of a body performing pure rolling on an inclined plane depends on its shape and moment of inertia. The general formula is:
a = [g sinθ] / [1 + (k^2/r^2)]
Where:
- g = acceleration due to gravity
- θ = angle of inclination
- k = radius of gyration (k^2 = I/m)
- r = radius of the object
For example:
- Solid sphere: a = (5/7)g sinθ
- Solid cylinder/disc: a = (2/3)g sinθ
- Ring: a = (1/2)g sinθ
3. What is the condition for pure rolling on an inclined plane?
The condition for pure rolling is that the linear velocity of the center of mass and angular velocity about the axis must be related by v = rω. For an incline:
- The point of contact has zero velocity relative to the surface.
- Sufficient friction is required to prevent slipping.
- If v ≠ rω, slipping occurs instead of pure rolling.
4. Does pure rolling mean that no slipping occurs?
Yes, pure rolling means there is no slipping between the rolling object and the surface. The velocity at the point of contact is zero with respect to the inclined plane at every instant.
5. What is the direction of friction during pure rolling on an inclined plane?
During pure rolling down an inclined plane, friction acts up the incline to prevent the rolling body from slipping. Key points:
- Friction is necessary for pure rolling, but if the plane is too slippery (frictionless), pure rolling cannot occur.
- If the object is forced to roll up the incline or decelerate, friction acts down the incline.
6. Can pure rolling happen on a frictionless inclined plane?
No, pure rolling cannot occur on a frictionless inclined plane because friction is essential to synchronize rotational and translational motions. Without friction, an object will slip instead of roll.
7. What is the work done by friction in pure rolling motion on an inclined plane?
The work done by friction in pure rolling on an inclined plane is zero. This is because the point of contact does not move with respect to the surface, so friction does not do any net work.
- All energy goes into translational and rotational kinetic energy.
8. Why does friction do no work during pure rolling?
In pure rolling, the point of contact between the rolling body and the surface remains stationary with respect to the surface at each instant, so the displacement at the point of contact is zero, and work done by friction is zero.
9. Is the acceleration the same for a ring, disc, and sphere in pure rolling?
No, the acceleration during pure rolling differs for a ring, disc, and sphere because each has a different moment of inertia. For the same incline:
- Ring: a = (1/2)g sinθ
- Disc: a = (2/3)g sinθ
- Solid sphere: a = (5/7)g sinθ
10. In what situation does friction act up or down the incline during rolling?
When a body rolls down an inclined plane, friction usually acts up the incline (opposing slipping). If the body is rolled upwards or decelerated, friction acts down the incline. The actual direction depends on whether it supports or opposes pure rolling.
11. How does mass distribution (moment of inertia) affect pure rolling acceleration?
The moment of inertia determines how much of the gravitational force goes into rotation versus translation. Objects with larger moments of inertia (like rings) have lower acceleration than those with smaller moments (like spheres) for the same incline. Formula: a = [g sinθ] / [1 + (I/mr²)].
12. How can you experimentally confirm pure rolling on an inclined plane?
To confirm pure rolling:
- Mark a point on the body (e.g., a wheel) and observe if it returns to the lowest point after each rotation.
- Check that the track is rough enough for no slipping.
- Measure v and ω; if v = rω, pure rolling is occurring.
