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τ=F_{t}d,
where d 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 counter clockwise rotation implies positive torque).
(b) What is the magnitude of the torque τ on the pole, about point A, due to the tension in the rope?
Express your answer in terms of T, L and θ.
The moment arm method involves finding the effective moment armof the force. To do this, imagine a line parallel to the force,running through the point at which the force is applied, andextending 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 the 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.

This problem can be solved using the moment arm and tangential force methods.
To solve for the tangential part of the force, first use the trigonometric functions.
To solve for tension force torque, you can use the torque equation from the tangential force method. Next, use the trigonometric functions to solve the moment arm.
Identify the best solution for your problem.
The tangential force method is based on finding the component perpendicular the displacement of the pivot point to the place where the force is applied. The tangential force is the perpendicular component.
The torque equation for the tangential force method uses is.
τ=F_{t}d
Here, F_{t }is the tangential force,an d is the distance from the pivot point to the point where force is applied.
Moment arm is a method of finding the effective moment arms of the force. The moment arm is the displacement.
The moment arm method uses the torque equation.
τ=FR_{m}
Here F, is the force, and R_{m} is the moment arm.
The trigonometric identity can be described as follows:
cos θ=adjacent/hypotenuse
Here, θ is the angle,addjacent is the side adjacent to the angle θ, and hypotenuse is the longest side of the rightangled triangle.
(a) To find the F_{t} tangential force, draw the diagram.
To solve for the Tangential component of the Tension force T, use the cosine function
Substitute T
cos θ=F_{t/}T
F_{t} =T cos θ
(b) Use the torque equation for tangential force method.
Substitute T cos θ for Ft, and L for d in the equation τ=F_{t}d
τ=(T cos θ) L
=T L cos θ
(c) Refer the diagram in the question with Rm. The Rm is the adjacent to the angle θ and L is the hypotenuse.
Use cosine function to solve for the Rm
Substitute L for hypotenuse, and Rm for adjacent in the equation
cos θ=adjacent/hypotenuse and rearrange to solve for Rm.
cos θ= Rm / L
Rm=L cos θ
Use the torque equation of moment arm method.
Substitute T for F and L cos θ for d in the equation τ=FR_{m}
τ=T L cos θ
(d) From part b and c, it is clear that using ether method,we get the same torque
τ=T L cos θ
Thus,for this part also either can be used as both tthe method leads to same answer.
Ans: Part A
The magnitude of the tangential force acting on the pole because of the tension in the rope is F_{t} = T cos θ
This problem can be solved using the moment arm and tangential force methods.
To solve for the tangential part of the force, first use the trigonometric functions.
To solve for tension force torque, you can use the torque equation from the tangential force method. Next, use the trigonometric functions to solve the moment arm.
Identify the best solution for your problem.
The tangential force method is based on finding the component perpendicular the displacement of the pivot point to the place where the force is applied. The tangential force is the perpendicular component.
The torque equation for the tangential force method uses is.
[katex]\tau = {F_{\rm{t}}}d[/katex]
Here, [katex]{F_{\rm{t}}}[/katex]
Moment arm is a method of finding the effective moment arms of the force. The moment arm is the displacement.
The moment arm method uses the torque equation.
[katex]\tau = F{R_{\rm{m}}}[/katex]
Here, [katex]F[/katex]
The trigonometric identity can be described as follows:
[katex]\cos \theta = \frac{{{\rm{adjacent}}}}{{{\rm{hypotenuse}}}}[/katex]
Here, [katex]\theta[/katex]
(a)
To find the [katex]{F_{\rm{t}}}[/katex] tangential force, draw the diagram.
To solve for the Tangential component of the Tension force [katex]T[/katex], use the cosine function
Substitute [katex]T[/katex]
[katex]\begin{array}{c}\\\cos \theta = \frac{{{F_{\rm{t}}}}}{T}\\\\{F_{\rm{t}}} = T\cos \theta \\\end{array}[/katex]
(b)
Use the torque equation for tangential force method.
Substitute [katex]{F_{\rm{t}}}[/katex]0
[katex]{F_{\rm{t}}}[/katex]1
(c)
Refer to the diagram in the [katex]{F_{\rm{t}}}[/katex]2 question
To solve the [katex]{F_{\rm{t}}}[/katex]2 problem, use cosine function
Substitute [katex]{F_{\rm{t}}}[/katex]4
[katex]{F_{\rm{t}}}[/katex]5
Use the torque equation method for moment arm.
Substitute [katex]T[/katex]
[katex]{F_{\rm{t}}}[/katex]7
(d)
Parts b and c show that we can get the same torque using either method
[katex]{F_{\rm{t}}}[/katex]7
This means that both methods can be used for this section, as they each lead to the same answer.
Ans: Part A
The magnitude of the tangential force acting on the pole because of the tension in the rope’s length is [katex]{F_{\rm{t}}}[/katex]9