Decompose the force vector F_vec into radial (i.e., parallel to r_vec) and tangential (perpendicular to r_vec) components as shown. Find the magnitude of the radial and tangential components, F_r and F_t. You may assume that theta is between zero and 90 degrees.

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the torque about point p is proportional to the length r of the position vector r.

Is the following statement true or false?

The moment about point P is proportional to the length r of the position vector, r

Both the radial and tangential components of F generate moments about point P.

In this problem, the tangential force vector would tend to rotate an object clockwise around pivot point P.

Find the moment M about the pivot point P due to the force F. Your answer should correctly express both the magnitude and sign of M.

Express your answer in terms of either Fr and r or F, ?, and r.

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The tangential force method involves finding the component of the applied force that is perpendicular to the displacement from the pivot point to where the force is applied. This perpendicular component of the force is called the tangential force.

(a) What is , the magnitude of the tangential force that acts on the pole due to the tension in the rope?

Express your answer in terms of

and

.

When using the tangential force method, you calculate the torque using the equation

$\tau = f_{\rm t} d$ ,

where

is the distance from the pivot to the point where the force is applied. The sign of the torque can be determined by checking which direction the tangential force would tend to cause the pole to rotate (where counterclockwise rotation implies positive torque).

(b) What is the magnitude of the torque

on the pole, about point A, due to the tension in the rope?

Express your answer in terms of

, and

The moment arm method involves finding the effective moment arm of the force. To do this, imagine a line parallel to the force, running through the point at which the force is applied, and extending off to infinity in either direction. You may shift the force vector anywhere you like along this line without changing the torque, provided you do not change the direction of the force vector as you shift it. It is generally most convenient to shift the force vector to a point where the displacement from it to the desired pivot point is perpendicular to its direction. This displacement is called the moment arm.

For example, consider the force due to tension acting on the pole. Shift the force vector to the left, so that it acts at a point directly above point A in the figure. The moment arm of the force is the distance between the pivot and the tail of the shifted force vector. The magnitude of the torque about the pivot is the product of the moment arm and force, and the sign of the torque is again determined by the sense of the rotation of the pole it would cause.