Yahoo 知識+ 將於 2021 年 5 月 4 日 (美國東岸時間) 停止服務。從 2021 年 4 月 20 日 (美國東岸時間) 起，Yahoo 知識+ 網站將轉為僅限瀏覽模式。其他 Yahoo 資產或服務，或你的 Yahoo 帳戶將不會有任何變更。你可以在此服務中心網頁進一步了解 Yahoo 知識+ 停止服務的事宜，以及了解如何下載你的資料。
circle rotate without friction
im arguing with my physic teacher about 'will a circle rotated without friction, or it will just slide down the hill'
..............and i believe it gonna rotated due to the torque on the circle......do you think im correct, please support your answer with evidence
thanks a lot ><
- ?Lv 71 十年前最愛解答
I suppose you ask whether a sphere could slide down an inclined plane without rotation if the surface of the plane is perfectly smooth.
There are several different scenarios:
1. The sphere is initally at rest on the smooth inclined plane, and it slides down by its own weight component that is parallel to the plane:.
In this situation, the weight component (which is an external force given by the earth) acts on the centre of mass of the sphere. The sphere would just slide down the plane without any rotation, as there is no torque acting on the sphere to make it rotate.
[If friction is present on the inclined plane, frictional force, which acts backward up the slope, would provide a torque about the centre of mass of the sphere to make it rotate].
2. An external force (with direction parallel to the plane) is given to the sphere with line of action through its centre of mass:
This is similar to the first scenario just described. The external force just adds to the weight component parallel to the plane. The sphere slides down the smooth inclined plane without any rotation.
3. An external force(with direction parallel to the plane) is given to the sphere with line of action at some distance from the centre of mass, e.g. applied at the top of the sphere:
Under this situation, there is an external torque given to the sphere, it would slide down the plane and rotate at the same time. But be aware that the rotation motion of the sphere and the linear motion of the centre of mass are completely independent. That is to say, if the force is applied at the top of the sphere parallel to the plane, the relationship:
velocity of centre of mass = radius of sphere x angular velocity of rotating sphere does not hold.
The rotational speed of the sphere only depends on the magnitude of the applied torque, and is independent on the linear speed of the sphere, as well as on the angle of slope.