# Rough horizontal surface and an inclined smooth plane

## Main Question or Discussion Point

An object rolls down a smooth plane (whose angle and length are given) starting from rest. At the end of the plane it reaches horizontal rough surface, it rolls again on this surface and comes to rest. Given the friction coefficient how do we calculate the distance travelled by the object on the horizontal surface.

We can calculate the object velocity at the end of the inclined plane (interface of the two surfaces). Now what happens? Should be consider for further calculation the speed of the object along the horizontal direction or the actual speed at the interface should be continued? And what are the reasons for this answer?

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Hootenanny
Staff Emeritus
Gold Member
Firstly, a body would not roll down a frictionless plane, it would slide. Friction is required for rolling. One can calculate the final speed at which a body will leave the incline plane fairly trivially using conservation of energy, or Newton's second law if one prefers. Once the particle reaches the rough horizontal surface one again needs to use Newton's second law to determine the acceleration of the body on that surface. Using this information one can then treat Newton's second law as an ODE and solve it for displacement with the initial condition that the body was traveling with the speed calculated at the bottom of the inclined plane.

Can you please explain why friction is needed for rolling, why without friction, it will not roll. I believe it is correct but I am unable to think myself why it is so. Thanks.

Hootenanny
Staff Emeritus
Gold Member
An excellent explanation is presented http://cnx.org/content/m14385/latest/" [Broken].

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Yes, it is excellent. Thanks for enlightening me.

for a body to roll you need a torque which is nothing but a turning effect produced when 2 forces whose lines of action donot coincide act on the same body but in different directions.When a body is placed on a surface one force can be offered by us,but the other has to be by friction

Shooting Star
Homework Helper
Once the particle reaches the rough horizontal surface one again needs to use Newton's second law to determine the acceleration of the body on that surface. Using this information one can then treat Newton's second law as an ODE and solve it for displacement with the initial condition that the body was traveling with the speed calculated at the bottom of the inclined plane.
Hi Hootenanny,

Technically, if the horizontal rough surface only provides sliding friction, but no rolling friction, then once the object has started to roll without slipping, there will be no deceleration of the body.

I mention this because I feel the OP should understand the three kinds of friction and when to apply them. For example, for a wheel rolling down a plane without slipping, it's the maximum value of static friction which is important.

Hootenanny
Staff Emeritus
Gold Member
Hi Hootenanny,

Technically, if the horizontal rough surface only provides sliding friction, but no rolling friction, then once the object has started to roll without slipping, there will be no deceleration of the body.

I mention this because I feel the OP should understand the three kinds of friction and when to apply them. For example, for a wheel rolling down a plane without slipping, it's the maximum value of static friction which is important.
Hi SS,

Indeed, if we assume that both the body and the surface are rigid and that there is no slipping (i.e. the velocity of the point on the body which is in contact with the surface is zero relative to the surface) then there will be no acceleration of the body.

The OP originally states that the plane is smooth and hence the body will slide down the plane with no angular velocity, the motion is purely translational. However, once the body reaches the rough horizontal surface it will most definitely accelerate, the body will undergo both a translational and rotational acceleration. The translational velocity will decrease and the rotational velocity will increase until the point at which the "no-slip" condition is satisfied, that is when the velocity of the point on the body which is in contact with the surface is zero relative to the surface. At this point there will be no acceleration and the body will continue to roll indefinitely. One must use Newton’s second law to describe the transition between purely translational and translational and rotational motion, which is what I was refering to in my OP. However as you correctly say, once the “no-slip” condition is satisfies there will be no further acceleration.