A force of magnitude , making an angle with the x axis, is appliedto a particle located at point A, at Cartesian coordinates (0, 0)in the figure. The vector and the four reference points (i.e., A,B, C, and D) all lie in the xy plane. Rotation axes A - D lieparallel to the z axis and pass through each respective referencepoint.
The torque of a force acting on a particle having a position vectorwith respect to a reference point (thus points from the referencepoint to the point at which the force acts) is equal to the crossproduct of and , . The magnitude of the torque is , where is theangle between and ; the direction of is perpendicular to both and .For this problem ; negative torque about a reference pointcorresponds to clockwise rotation. You must express in terms of , ,and/or when entering your answers.
A force of magnitude , making an angle with the x axis, is appliedto a particle located at point A, at Cartesian coordinates (0, 0)in the figure. The vector and the four reference points (i.e., A,B, C, and D) all lie in the xy plane. Rotation axes A - D lieparallel to the z axis and pass through each respective referencepoint.
The torque of a force acting on a particle having a position vectorwith respect to a reference point (thus points from the referencepoint to the point at which the force acts) is equal to the crossproduct of and , . The magnitude of the torque is , where is theangle between and ; the direction of is perpendicular to both and .For this problem ; negative torque about a reference pointcorresponds to clockwise rotation. You must express in terms of , ,and/or when entering your answers.
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A forceĀ FāĀ Ā of magnitudeĀ FĀ making an angleĀ ĪøĀ 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 FāĀ Ā lies in theĀ XYĀ plane, and the four axes of rotation A, B, C, and D all lie perpendicular to theĀ XYĀ plane. (Figure 1)
A particle is located at a vector positionĀ rāĀ Ā with respect to an axis of rotation (thusĀ rāĀ Ā points from the axis to the point at which the particle is located). The magnitude of the torqueĀ ĻĀ about this axis due to a forceĀ Fā acting on the particle is given by
Ļ=rFsin(Ī±)=(moment arm)ā F=rFā„,
whereĀ Ī±Ā is the angle betweenĀ rāĀ Ā andĀ Fā ,Ā rĀ is the magnitude ofĀ rā ,Ā FĀ is the magnitude ofĀ Fā , the component ofĀ rāĀ Ā that is perpendicular to FāĀ Ā is the moment arm, andĀ Fā„ is the component of the force that is perpendicular toĀ rā .
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 angleĀ Ī±Ā in the above equation in terms ofĀ Īø,Ā Ļ, and/orĀ ĻĀ when entering your answers. Keep in mind thatĀ Ļ=180degrees and (Ļ/2)=90 degrees.
Part A
What is the torqueĀ ĻA about axis A due to the forceĀ Fā ?
Express the torque about axis A at Cartesian coordinates (0,0).
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| ĻA = | Ā |
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Part B
What is the torqueĀ ĻB about axis B due to the forceĀ Fā ? (B is the point at Cartesian coordinates (0,b), located a distanceĀ bĀ from the origin along theĀ yĀ axis.)
Express the torque about axis B in terms ofĀ F,Ā Īø,Ā Ļ,Ā Ļ, and/or other given coordinate data.
| Ā |
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| ĻB = | Ā |
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Part C
What is the torqueĀ ĻC about axis C due toĀ Fā ? (C is the point at Cartesian coordinates (c,0), a distanceĀ cĀ along theĀ xĀ axis.)
Express the torque about axis C in terms ofĀ F,Ā Īø,Ā Ļ,Ā Ļ, and/or other given coordinate data.
| Ā |
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| ĻC = | Ā |
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Part D
What is the torqueĀ ĻD about axis D due toĀ Fā ? (D is the point located at a distanceĀ dĀ from the origin and making an angleĀ ĻĀ with theĀ xĀ axis.)
Express the torque about axis D in terms ofĀ F,Ā Īø,Ā Ļ,Ā Ļ, and/or other given coordinate data.
