Question

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

(a) What is $F_t$ , the magnitude of the tangential forcethat 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 torqueusing the equation

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

where is the distance from the pivot tothe point where the force is applied. The sign of the torque can bedetermined by checking which direction the tangential force wouldtend to cause the pole to rotate (where counterclockwise rotationimplies positive torque).

(b) What is the magnitude of the torque on the pole, about point A, due to thetension in the rope?

Express your answer in terms of, , and .

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 theforce vector anywhere you like along this line without changing thetorque, provided you do not change the direction of the forcevector as you shift it. It is generally most convenient to shiftthe force vector to a point where the displacement from it to thedesired pivot point is perpendicular to its direction. Thisdisplacement 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 pointdirectly above the point A in the figure. The moment arm of theforce is the distance between the pivot and the tail of the shiftedforce vector. The magnitude of the torque about the pivot is theproduct of the moment arm and force, and the sign of the torque isagain determined by the sense of the rotation of the pole it wouldcause.

(c) Find $R_m$ , the length of the moment arm of theforce.

Express your answer in terms of and .

Now consider a woman standing on the ball of her foot as shown. Anormal force of magnitude acts upward on the ball of her foot. TheAchilles' tendon is attached to the back of the foot. The tendonpulls on the small bone in the rear of the foot with a force. This small bone has a length , and the angle between this bone and theAchilles' tendon is . The horizontal displacement between theball of the foot and the point P is .

(d) Suppose you were asked to find the torqueabout point P due to the force of magnitude in the Achilles' tendon. Which of the followingstatements is correct?

Thetangential force method must be used. The momentarm method must be used. Eithermethod may be used. Neithermethod can be used.

Answer 1

Concepts and reason

The method required to solve this problem are tangential force method and moment arm method.

Initially use the trigonometric function to solve for the tangential component of the force.

Later use the torque equation used in the tangential force method to solve for torque due to tension force. Then use the trigonometric function to solve for the moment arm.

Finally identify the method that can be used for the given problem.

Fundamentals

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.

The torque equation used in the tangential force method is,

$\tau = {F_{\rm{t}}}d$

Here, ${F_{\rm{t}}}$ is the tangential force, and d is the distance from the pivot point to the point where force is applied.

The moment arm method involves finding the effective moment arm of the force. The displacement is called the moment arm.

The torque equation used in the moment arm method is,

$\tau = F{R_{\rm{m}}}$

Here, $F$ is the force, and ${R_{\rm{m}}}$ is the moment arm.

The trigonometric identity is as follows:

$\cos \theta = \frac{{{\rm{adjacent}}}}{{{\rm{hypotenuse}}}}$

Here, $\theta$ is the angle, adjacent is the side adjacent to the angle $\theta$ , and hypotenuse is the longest side of the right-angled triangle.

(a)

Draw the sketch of the diagram to find the tangential force ${F_{\rm{t}}}$ . The Tension in the rope is $T$ . The angle between the tension and tangential force is $\theta$ .

Use cosine function to solve for the tangential component of the Tension force $T$ .

Substitute $T$ for ${\rm{hypotenuse}}$ , and ${F_{\rm{t}}}$ for adjacent in the equation $\cos \theta = \frac{{{\rm{adjacent}}}}{{{\rm{hypotenuse}}}}$ and rearrange to solve for ${F_{\rm{t}}}$ .

$\begin{array}{c}\\\cos \theta = \frac{{{F_{\rm{t}}}}}{T}\\\\{F_{\rm{t}}} = T\cos \theta \\\end{array}$

(b)

Use the torque equation of tangential force method.

Substitute $T\cos \theta$ for ${F_{\rm{t}}}$ , and $L$ for $d$ in the equation $\tau = {F_{\rm{t}}}d$ .

$\begin{array}{c}\\\tau = \left( {T\cos \theta } \right)L\\\\ = TL\cos \theta \\\end{array}$

(c)

Refer the diagram in the question with ${R_{\rm{m}}}$ . The ${R_{\rm{m}}}$ is the adjacent to the angle $\theta$ and $L$ is the hypotenuse.

Use cosine function to solve for the ${R_{\rm{m}}}$ .

Substitute $L$ for ${\rm{hypotenuse}}$ , and ${R_{\rm{m}}}$ for adjacent in the equation $\cos \theta = \frac{{{\rm{adjacent}}}}{{{\rm{hypotenuse}}}}$ and rearrange to solve for ${R_{\rm{m}}}$ .

$\begin{array}{c}\\\cos \theta = \frac{{{R_{\rm{m}}}}}{L}\\\\{R_{\rm{m}}} = L\cos \theta \\\end{array}$

Use the torque equation of moment arm method.

Substitute $T$ for $F$ , and $L\cos \theta$ for $d$ in the equation $\tau = F{R_{\rm{m}}}$ .

$\tau = TL\cos \theta$

(d)

From part b and c, it is clear that using either method, we get the same torque

$\tau = TL\cos \theta$

Thus, for this part also either method can be used as both the method leads to same answer.

Ans: Part a

The magnitude of tangential force that acts on the pole due to the tension in the rope is ${F_{\rm{t}}} = T\cos \theta$ .