# Help Solving 2 Qs: Tan(Ɵ) & Sign Direction Error

• gnits
In summary, the back of the book says that if you equate forces vertically for the vertical rod only you get:F1 = W/2If you equate forces for the whole system vertically you get:R = (3/2)WHowever, if you take clockwise moments about C for rod CD only you get:F * L sin(t) + W * (L/2) cos(t) = R * L cos(t)Substituting for R and rearranging gives: F = W / tan(t)
gnits
Homework Statement
To find possible angle of two leaning rods
Relevant Equations
moments

The last part follows easily from the first part.

Answer from back of book for first part is:

2/(3u') <= tan(Ɵ) <= 2u

What I have done is the following:

Here's my diagram (I have separated the components to show the internal forces in the system. I have used t instead of Ɵ) :

Orange forces are internal forces.

If I equate forces for the whole system vertically I get:

R = (3/2)W

If I equate forces vertically for the vertical rod only I get:

F1 = W/2

If I take clockwise moments about C for rod CD only I get:

F * L sin(t) + W * (L/2) cos(t) = R * L cos(t)

Substituting for R and rearranging gives: F = W / tan(t)

Now for no slipping at D we need F <= u' R so this gives:

W/tan(t) <= u' * (3/2)W

Which leads to tan(t) <= 2/(3u') which is the answer asked for but with the sign reversed.

1) How have I gotten the sign mixed up?

Taking clockwise moments about D for rod CD only I get:

S * L sin(t) = F1 * L cos(t) + W * (L/2) * cos*(t)

Substituting for F1and rearranging gives:

S = W/tan(t) and for no slipping at C we need S <= u * F1

So gives W/tan(t) <= u * W/2 which leads to tan(t) >= 2/u

Again wrong direction of sign and also wrong answer as we need tan(t) <= 2u

2) Where have a erred?

Thanks for any help.

So, notwithstanding the signs, would you (or others) think my analysis of taking moments for CD only about D looks correct? I ask because here I differ in the actual form of the answer. The given answer is 2u but I arrive at 2/u. If you\others agree that 2/u is correct then I would put it down to a typo in the book.

Thanks again.

Last edited:
gnits said:
Now for no slipping at D we need F <= u' R so this gives:

W/tan(t) <= u' * (3/2)W

Which leads to tan(t) <= 2/(3u') which is the answer asked for but with the sign reversed.
I agree with the first equation here, but that implies that ##\mu' \tan \theta \ge \dfrac 2 3##.

PeroK said:
I agree with the first equation here, but that implies that ##\mu' \tan \theta \ge \dfrac 2 3##.
Yes indeed, I agree. I had fallen at the last there. Thanks for that. So only the second part is now causing me issues.

gnits said:
S = W/tan(t) and for no slipping at C we need S <= u * F1
For no slipping we need ##\mu S \ge F_1 = \dfrac{mg}{2}##.

##S## is the normal force and ##F_1## is the required friction force.

PeroK said:
For no slipping we need ##\mu S \ge F_1 = \dfrac{mg}{2}##.

##S## is the normal force and ##F_1## is the required friction force.
Thanks very much indeed. I had swapped S and F1. I had written S <= u * F1 whereas, as you say, I should have written F1 <= u S. And indeed this leads to the book's answer. Thank you very much.

gnits said:

So, notwithstanding the signs, would you (or others) think my analysis of taking moments for CD only about D looks correct? I ask because here I differ in the actual form of the answer. The given answer is 2u but I arrive at 2/u. If you\others agree that 2/u is correct then I would put it down to a typo in the book.

Thanks again.
Sorry, I deleted my reply to write a better one, but then I had an emergency that kept me away from the task. You must have read my post soon after I posted it but before I deleted it. Anyway, it looks like your issue has been resolved so there is nothing more for me to say. I apologize for the confusion.

## 1. What is the formula for finding the tangent of an angle?

The formula for finding the tangent of an angle is tan(Ɵ) = opposite/adjacent. This means that to find the tangent, you divide the length of the side opposite the angle by the length of the side adjacent to the angle.

## 2. How do I solve for the tangent of an angle?

To solve for the tangent of an angle, you need to know the length of the opposite and adjacent sides. Once you have those values, you can plug them into the formula tan(Ɵ) = opposite/adjacent and calculate the tangent. Make sure to use a calculator or table to find the actual value of the tangent.

## 3. What does a positive or negative tangent value indicate?

A positive tangent value indicates that the angle is in the first or third quadrant of the unit circle, while a negative value indicates that the angle is in the second or fourth quadrant. This can also be interpreted as the direction of the angle's rotation: positive for counterclockwise and negative for clockwise.

## 4. How do I determine the sign direction error in my tangent calculation?

To determine the sign direction error in your tangent calculation, first find the actual value of the tangent using a calculator or table. Then, compare it to your calculated value. If they have the same sign, there is no error. If they have different signs, the error is in the sign direction.

## 5. How can I avoid sign direction errors when calculating tangents?

To avoid sign direction errors when calculating tangents, make sure to use the correct quadrant when finding the lengths of the opposite and adjacent sides. Also, double check your calculations and use a calculator or table to find the actual value of the tangent to compare with your calculated value.

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