# Tension in a rotating rod at various places

• brotherbobby
In summary, the book's answer states that ##\boxed{T_1\; >\; T_2}##, but based on the derived centripetal forces at points ##L/4## and ##3L/4##, it can be concluded that ##T_2 > T_1##. This is due to the fact that the tension at point ##3L/4## is not only rotating a point mass, but also providing the force to rotate the entire length of the rod. Taking into account the entire body from the point of interest to the outer end, the sum of forces will result in ##T_2 > T_1##.
brotherbobby
Homework Statement
A rod of length ##L## is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let ##T_1## and ##T_2## be the tensions at the points ##\frac{L}{4}## and ##\frac{3L}{4}## (respectively) away from the pivoted ends. Compare the magnitudes of ##T_1## and ##T_2##.
Relevant Equations
Centripetal force needed to keep a particle of ##m## moving in a circle of radius ##r## with angular velocity ##\omega##: ##\;\;\mathbf{F_C = m\omega^2r}##.
(The answer given in the text says ##\boxed{T_1\; >\; T_2}## but, as I show below, I think it's just the opposite).

I begin by putting an image relevant to the problem above. Taking a small particle each of the same mass ##m## at the two positions, the centripetal forces are ##T_1 = \frac{m\omega^2 L}{4}## and ##T_2 = \frac{3m\omega^2 L}{4}##.

Clearly, from above, we have ##T_2 > T_1##, contrary to the answer given in the book.

simi simran
You got your masses swapped. The mass to the right of points 1 and 2 is what counts. At 3L/4 there is less mass being accelerated by the tension.

brotherbobby said:
Homework Statement: A rod of length ##L## is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let ##T_1## and ##T_2## be the tensions at the points ##\frac{L}{4}## and ##\frac{3L}{4}## (respectively) away from the pivoted ends. Compare the magnitudes of ##T_1## and ##T_2##.
Homework Equations: Centripetal force needed to keep a particle of ##m## moving in a circle of radius ##r## with angular velocity ##\omega##: ##\;\;\mathbf{F_C = m\omega^2r}##.

(The answer given in the text says ##\boxed{T_1\; >\; T_2}## but, as I show below, I think it's just the opposite).

View attachment 250853

I begin by putting an image relevant to the problem above. Taking a small particle each of the same mass ##m## at the two positions, the centripetal forces are ##T_1 = \frac{m\omega^2 L}{4}## and ##T_2 = \frac{3m\omega^2 L}{4}##.

Clearly, from above, we have ##T_2 > T_1##, contrary to the answer given in the book.

The tension at the point ##L/4## is not simply rotating a point mass there. It's providing the force to rotate the rest of the bar, all the way out to ##L##.

It is not a single particle at each location that is relevant. It is the body extending from the point in question to the outer free end that needs to be considered. Draw a FBD for the outer end of the rod from a point of interest to the outboard end. Then sum forces on that FBD and that will get the answer you were shown.

## 1. What is tension in a rotating rod?

Tension in a rotating rod is the force that is exerted on the rod as it rotates. It is caused by the centripetal force that is required to keep the rod moving in a curved path.

## 2. How does tension vary at different places along the rod?

Tension in a rotating rod varies depending on the distance from the center of rotation. The tension increases as you move away from the center and towards the ends of the rod.

## 3. What factors affect the tension in a rotating rod?

The tension in a rotating rod is affected by the speed of rotation, the mass of the rod, and the distance from the center of rotation.

## 4. How is the tension in a rotating rod related to its angular velocity?

The tension in a rotating rod is directly proportional to its angular velocity. This means that as the angular velocity increases, so does the tension in the rod.

## 5. Can tension in a rotating rod cause deformation?

Yes, if the tension in a rotating rod becomes too great, it can cause the rod to deform or even break. This is why it is important to consider the maximum tension that a rod can withstand when designing rotating systems.

Replies
31
Views
1K
Replies
18
Views
363
Replies
3
Views
418
Replies
40
Views
3K
Replies
25
Views
822
Replies
39
Views
5K
Replies
38
Views
2K
Replies
10
Views
1K
Replies
5
Views
689
Replies
9
Views
2K