Tension in a rope at the midpoint

In summary, the tension (Tq) at the midpoint (P) of a massless rope is calculated to be 50N, while the tension (Tp) at the same point on a rope with mass is found to be either 20N or 30N. After further analysis, it is concluded that Tp = 30N is the correct value, as the system should only consider the forces acting on it. Additionally, the tension in a hanging system does not depend on the mass above or below, but rather on the forces acting on the system. Considering all forces in the system, the correct equation for Tp is Tp - Tq - 0.1g - 1.9g = 2
  • #1
James2911
13
0

Homework Statement



Calculate Tension (Tp) and (TQ) (P is the mid point)
ck_5a014f1d21c72.png


Homework Equations



T - mg = ma

The Attempt at a Solution


For Q (massless rope)
Tq - 1.9g - 0.2g - 2.9g = 5a
Tq = 5(g+a)
Tq = 5 * 10 = 50N

Given answer = 50N

(Calculation of Tension for Q is clear to me, but I am getting 2 answers for tension of P)

Answer 1 - For P (rope with mass, hence tension can't be considered equal on all points)
Tp - .1g - 2.9g = 3a
Tp = 3(g+a) = 30N

Answer 2 - For P
Tp - 1.9g - 0.1g = 2a
Tp = 2(g+a) = 20N

The given answer is Tp = 30N
Now here's my doubt.

In answer 1 - we are calculating Tension at point P. We considered the weights 0.1kg and 2.9kg and ignored that the 1.9kg weight. But I think the weight 1.9kg would apply a force 1.9g which should be considered.(Here we come up from down and include one weight (1.9kg and .1kg)

In answer 2 - we considered the weights 1.9kg and 0.1kg but didn't include 2.9kg. So, here we ignored 2.9kg like it didn't exist. But wouldn't the 2.9g exert a force? (Here we go up from down and include 2.9kg and 0.1kg(half of mass of rope)

--------------------------------------------------------------------------------------------------------------------
Out of answer 1 and 2, I consider Answer 2 i.e., 20N to be more right than the second one because, we are asked to find Tension at the midpoint. So we should just only consider the forces that are acting on it.

Please help. I've tried to find more solutions of this question online. Tq always comes out to be 50N.
But the value of Tp seems to be either 20N or 30N on different forums. Which is the correct value?

(reiteration: Tp = 30N is the given answer)
 

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  • #2
What free body diagrams are you considering in the two different cases? Remember to include all forces acting on the system, this includes all tensile forces.

Edit: Until you feel more secure: Always draw free body diagrams.
 
  • #3
Your answer 1 is correct. Your force balance for answer 2 is incorrect.
 
  • #4
James2911 said:

Homework Statement



Calculate Tension (Tp) and (TQ) (P is the mid point)
View attachment 238043

Homework Equations



T - mg = ma

The Attempt at a Solution


For Q (massless rope)
Tq - 1.9g - 0.2g - 2.9g = 5a
Tq = 5(g+a)
Tq = 5 * 10 = 50N

Given answer = 50N

(Calculation of Tension for Q is clear to me, but I am getting 2 answers for tension of P)

Answer 1 - For P (rope with mass, hence tension can't be considered equal on all points)
Tp - .1g - 2.9g = 3a
Tp = 3(g+a) = 30N

Answer 2 - For P
Tp - 1.9g - 0.1g = 2a
Tp = 2(g+a) = 20N

The given answer is Tp = 30N
Now here's my doubt.

In answer 1 - we are calculating Tension at point P. We considered the weights 0.1kg and 2.9kg and ignored that the 1.9kg weight. But I think the weight 1.9kg would apply a force 1.9g which should be considered.(Here we come up from down and include one weight (1.9kg and .1kg)

In answer 2 - we considered the weights 1.9kg and 0.1kg but didn't include 2.9kg. So, here we ignored 2.9kg like it didn't exist. But wouldn't the 2.9g exert a force? (Here we go up from down and include 2.9kg and 0.1kg(half of mass of rope)

--------------------------------------------------------------------------------------------------------------------
Out of answer 1 and 2, I consider Answer 2 i.e., 20N to be more right than the second one because, we are asked to find Tension at the midpoint. So we should just only consider the forces that are acting on it.

Please help. I've tried to find more solutions of this question online. Tq always comes out to be 50N.
But the value of Tp seems to be either 20N or 30N on different forums. Which is the correct value?

(reiteration: Tp = 30N is the given answer)

In general, does the tension in a hanging system depend on the mass above or mass below?

It seems to me that if the tension at P is 20N, then shouldn't the tension at Q be 0N?
 
  • #5
James2911 said:
Answer 2 - For P
Tp - 1.9g - 0.1g = 2a
Tp = 2(g+a) = 20N
You appear to be analysing the system between P and Q to arrive at this. But that system also has the tension at Q acting, so that should feature in the equation. Also, some signs are wrong.
 
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Likes Chestermiller
  • #6
Guys, the OP was last seen at 2:28 pm viewing this thread, so he has seen all posts except haruspex's post. No response on his part.
 
  • #7
haruspex said:
You appear to be analysing the system between P and Q to arrive at this. But that system also has the tension at Q acting, so that should feature in the equation. Also, some signs are wrong.

Thank you! It worked!

if we consider the system between Q and P;
Tq - Tp - 0.1g - 1.9g = 2a
50 - 2(9.8+0.2) = Tp
Tp = 50 - 20
Tp = 30

This did bring out the right answer. But instead of my traditional method of drawing FBD of a body, I considered a system between P and Q and took the forces acting on it (Like Tq acting upwards, and the rest 3 acting downwards).

But, how do I relate the Tension Q to FBD of P? There is a block (1.9kg) in between.

Chestermiller said:
Guys, the OP was last seen at 2:28 pm viewing this thread, so he has seen all posts except haruspex's post. No response on his part.
I apologize.
 
  • #8
James2911 said:
But, how do I relate the Tension Q to FBD of P?
There is not such a thing as a FBD of a point. You need to consider a system that you draw a contour around. Your reaction here seems to suggest that you need to revisit FBDs and how they are used.

The FBD is a FBD including everything between P and Q.
 
  • #9
Orodruin said:
There is not such a thing as a FBD of a point. You need to consider a system that you draw a contour around. Your reaction here seems to suggest that you need to revisit FBDs and how they are used.

The FBD is a FBD including everything between P and Q.

Thank you sir!
 

Related to Tension in a rope at the midpoint

1. What is tension in a rope at the midpoint?

Tension in a rope at the midpoint is the amount of force exerted on the rope at its center point. It is the result of the weight of the rope and any additional forces acting on it.

2. How is tension in a rope at the midpoint calculated?

Tension in a rope at the midpoint can be calculated by dividing the total weight of the rope by two, as the tension is equal on both sides of the midpoint. Additional forces, such as pulling or stretching the rope, can also affect the tension at the midpoint.

3. Why is tension in a rope at the midpoint important to understand?

Understanding tension in a rope at the midpoint is important because it affects the stability and strength of the rope. If the tension is too high, the rope may break or become unstable. If the tension is too low, the rope may not be able to support the weight it is carrying.

4. How does the material and thickness of a rope affect tension at the midpoint?

The material and thickness of a rope can greatly impact the tension at the midpoint. Thicker and stronger materials, such as steel, can withstand higher tensions, while thinner and weaker materials, such as nylon, may have lower tension limits. The weight of the material also affects the overall weight of the rope and therefore the tension at the midpoint.

5. Can tension in a rope at the midpoint be changed?

Yes, tension in a rope at the midpoint can be changed by adjusting the weight of the rope or applying additional forces to it. For example, pulling on the rope will increase the tension, while reducing the weight of the rope will decrease the tension. It is important to carefully consider and monitor the tension in a rope at the midpoint to ensure safety and stability.

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