A force F⃗ of magnitude F making an angleθ with the x axis is applied to a particlelocated along axis of rotation A, at Cartesian coordinates (0,0) inthe figure. The vector F⃗ lies in the xy plane,and the four axes of rotation A, B, C, and D all lie perpendicularto the xy plane. (Figure 1)

A particle is located at a vector position r⃗ withrespect to an axis of rotation (thus r⃗ points from theaxis to the point at which the particle is located). The magnitudeof the torque τ about this axis due to a force F⃗acting on the particle is given by

τ=rFsin(α)=(momentarm)⋅F=rF⊥,

where α is the angle between r⃗ andF⃗ , r is the magnitude of r⃗ ,F is the magnitude of F⃗ , the component ofr⃗ that is perpendicualr to F⃗ is the moment arm,and F⊥ is the component of the force that is perpendicularto r⃗ .

Sign convention: You will need to determine thesign by analyzing the direction of the rotation that the torquewould tend to produce. Recall that negative torque about an axiscorresponds to clockwise rotation.

In this problem, you must express the angle α in theabove equation in terms of θ , ϕ , and/orπ when entering your answers. Keep in mind thatπ=180degrees and (π/2)=90degrees

Part A

What is the torque τA about axis A due to the forceF⃗ ?

Part B

What is the torque τB about axis B due to the forceF⃗ ? (B is the point at Cartesian coordinates(0,b) , located a distance b from the originalong the y axis.)

Part C

What is the torque τC about axis C due to F⃗ ?(C is the point at Cartesian coordinates (c,0) , adistance c along the x axis.)

Part D

What is the torque τD about axis D due to F⃗ ?(D is the point located at a distance d from the originand making an angle ϕ with the x axis.)