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A force F⃗  of magnitude F making an angle θ with the x-axis is applied to a particle located along the axis of rotation A, at Cartesian coordinates (0,0) in the figure. The vector F⃗  lies in the XY plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the XY plane. (Figure 1)

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A particle is located at a vector position r⃗  with respect to an axis of rotation (thus r⃗  points from the axis to the point at which the particle is located). The magnitude of the torque τ about this axis due to a force F⃗ acting on the particle is given by

τ=rFsin(α)=(moment arm)⋅F=rF⊥,

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

Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation.

In this problem, you must express the angle α in the above equation in terms of θϕ, and/or π when entering your answers. Keep in mind that π=180degrees and (π/2)=90 degrees.

Part A

What is the torque τA about axis A due to the force F⃗ ?

Express the torque about axis A at Cartesian coordinates (0,0).

 
   
τA =  

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Part B

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

Express the torque about axis B in terms of Fθϕπ, and/or other given coordinate data.

 
   
τB =  

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Part C

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

Express the torque about axis C in terms of Fθϕπ, and/or other given coordinate data.

 
   
τC =  

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Part D

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

Express the torque about axis D in terms of Fθϕπ, and/or other given coordinate data.

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