Rolling Without Slipping Down an Incline - Sphere, Disk, and Ring on a Ramp Using Forces and Torque

We will use forces and torque to learn why it is that if a solid sphere, disc, and hoop are rolled down a ramp, the sphere will win, the disc will come in second, and the hoop will finish last. This is independent of both the mass M and the radius R of the shapes. This is a classic problem in rotational mechanics. The problem can also be solved using energy considerations, but we take an approach using forces and torques here. 0:30 Draw Forces 1:00 Sum Forces. 2:00 Sum Torques. 3:12 Algebra. 3:40 General Result. 4:40 Comparing Shapes

[@minhajtanim7646]

As the director of torque due to friction force is clockwise means negative why there is no negative sign front of Fs.R?

[@dr.piercesphysicsmath9071]

This would cause a problem when trying to connect the rotation and translation. I have (arbitrarily) defined "down the ramp" as my + direction for translation. This means that translational accel and rotational accel (angular accel) will be positive as the object gains translational and rotational speed. The signs must agree if we wish to use a=R alpha or v = R omega to connect translation and rotation.
Your question is a great one, but holding too firmly to the clockwise/counterclockwise standard will cause you problems in this case unless you also adjust your + direction for translation.