Sign of torque when rolling an object down an incline Suppose you have an object rolling down the incline at 30 degrees.
 
Given the point of contact is instantaneously at rest, I decided to analyse torques at that point.  Therefore, the only force creating a torque is the force of gravity, here being 
$$
F = m g \sin(\theta) \Rightarrow \text{Torque} = I(\alpha) = m g \sin(\theta) R.
$$
Using both the right-hand rule and the fact that angular velocity is clockwise, torque should be negative. But the expression obtained suggests a positive torque. What am I missing here? Also, if the torque is negative, then the tangential acceleration should be negative too, but it makes sense to say that tangential acceleration is positive. Can someone please clarify these two points?
 A: You have not considered the direction of torque in you equation.
Since the Torque is caused by Frictional Force $F_r$ which is in the direction $-i$. the torque is $\vec{\tau}= \vec{r}\times\vec{F_r}$ and $\vec{r} $ is in $j$ direction
So the cross product yields
$\vec{\tau}= -|r||F_r|\hat{k}$
which ofcourse is in the negative direction of positive z-axis
A: @vishwaas is correct, but here is an explanation without vector math. By convention, $x$-$y$-$z$ ordinates obey their own RHR such that rotating an imaginary right hand from positive $x$-axis to positive $y$-axis around the $z$-axis (counter-clockwise) puts positive $z$ in the towards-the-observer direction (out of the page). The rotation of, and the torque on, the rolling object is clockwise, putting the direction of the torque vector (by RHR) away from the observer (into the page). So the torque vector is negative in the $z$ direction.
A: the method i use for determining the direction is quite simple. basically when you are to find the direction of a cross product say a x b just put your hand on a such that it is facing the b vector. thus for finding the direction of torque put your hand on the radius ( in the direction perpendicular to the force) such that it faces the force and curl your fingers the direction of curl will give you the direction of rotation.
