Rolling on an Incline: Will a Sphere Continue Pure Rolling?

[sachin123]

Homework Statement

A smooth inclined plane with inclination \theta is fixed in a car accelerating on a horizontal surface with a=gtan(\theta).A sphere is set pure rolling on this incline.
Will it continue pure rolling?

The Attempt at a Solution

My idea is to find the linear acceleration of the sphere using Newton's laws along the incline.

Then I find the angular acceleration equating Torque exerted by friction to the product of Moment of Inertia of the sphere and its angular acceleration.Knowing the angular acceleration,I find the linear acceleration by multiplying it with radius of the bob.
If the two accelerations are the same,then it is pure rolling.

Am I right?
When I did it this way,I found that it will not roll purely.

 

[jeppetrost]

As I understand, smooth means no friction? Not that it makes a difference, actually.
Have you drawn a free-body-diagram of the situation? You can make an accelerating system and give the sphere an acceleration oppiste of the one the system has (like, if you lay a ball in a bus and the bus accelerates, what you see inside the bus is the ball accelerating, but backwards).
Now, see if the sphere has any linear acceleration (in direction of the inclined plane). If it does, it will start to move either slower or faster than what it would be if it was 'pure rolling'. But if it has no acceleration, it will just keep on moving, pure rolling.
BTW, I got that it has no acceleration, and hence "yes, it will continue pure rolling.
Please write, if my explanation was confusing, and I'll draw the free-body-diagram.

 

[sachin123]

Hey.I actually missed that part of 'smooth'.
So,the bigger question I get is,how will it roll at all,if there isn't any friction?
I did that free diagram part as you said,and got 0 acceleration(linear).
So isn't it just supposed to slide down the incline with constant velocity?
Thanks.

 

[sachin123]

Where did I go wrong?

 

[jeppetrost]

Well, the problem states, that the sphere is "set pure rolling", so the rolling is started by "you" and not the incline. So, say it starts with some velocity v, then it has an angular velocity v/r, and when its velocity doesn't change (the 0 acceleration), it must continue pure rolling.