Find ` ∫ 7 cos⁡mx dx.`

Explanation

a. the forces on the pulley are ure.
i. attraction by the earth Mg vertically downward
ii tension tl by the rope along the rope
iii contact force N by the support at the centre.
`N=T+Mg` (centre of mass of the pulley is at rest so Newoton's 1st law applies.
b. The axis of rotation is the line through the centre of the pulley and perpendicular to theplane of the pulley.
c. Let us take the positive direction of the axis towards the reader.
The force Mg passses through the centre of mass and it intersects the axis of rotation. Hence the torque of Mg abut teh axils is zero (casse II). Similarly the torque of the contct force N is also zero.
The tension T is along the tangent of the rim in the vertically downward direction. The tension and the axis of rotation are perpendicular but never intersect. Case III applies. Join the point where the rope leaves the rim to the centre. TGhis line is the common perpendicular tot the tension and the axis. Hence the torque is T.r (positive, since it wil try to rotate the ulley anticlockwise).