Two objects joined by a rectilinear cable rotating

In summary, the scenario involves two objects connected by a straight cable that rotates around a fixed axis. The dynamics of the system are influenced by the tension in the cable, the mass of the objects, and their rotational motion, leading to various physical phenomena such as centripetal force and angular momentum. Understanding the mechanics of this setup can provide insights into rotational dynamics and the behavior of connected bodies in motion.
  • #1
Guillem_dlc
188
17
Homework Statement
We have two objects of mass ##3\, \textrm{kg}## joined by a rectilinear cable of ##3\, \textrm{m}## and negligible mass. The axis of rotation is normal to the cable and passes through it ##1\, \textrm{m}## from one of the objects, the moment of inertia of this object being ##7\, \textrm{kg m}^2##. If we want ##\omega =6\, \textrm{rad}/\textrm{s}## and ##L=10\, \textrm{kg m}^2/\textrm{s}##, what must be the moment of inertia of the second object with respect to the axis passing through its center of mass and parallel to the axis of rotation? Hints: It is advisable to draw a schematic of the system described. In addition, Steiner's theorem must be used in the calculations. Sol: ##15,67\, \textrm{kg m}^2##.
Relevant Equations
##L=I\omega##, Steiner's theorem
I've tried the following, but I don't get the correct result:

The moment of inertia of the system with respect to the axis of rotation is:

$$L=I\omega \Rightarrow I=\dfrac{L}{\omega}=\dfrac53 \, \textrm{kg m}^2$$

Then,

$$I=I_1+I_2\Rightarrow I_2=I-I_1=\dfrac53 -7=-\dfrac{16}3\, \textrm{kg m}^2$$

Finally, applying the Steiner's theorem:

$$I_2=I_{c2}+m_2d_2^2 \Rightarrow -\dfrac{16}{3}=I_{c2}+3\cdot 2^2 \Rightarrow I_{c2}=-17,33\, \textrm{kg m}^2$$

What have I done wrong in my reasoning?
 
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  • #2
Guillem_dlc said:
Hints: It is advisable to draw a schematic of the system described.
That sounds like a good hint. :wink: Can you upload your sketch please? Thanks.
 
  • #3
berkeman said:
That sounds like a good hint. :wink: Can you upload your sketch please? Thanks.
The problem description makes no sense to me. I cannot come up with a diagram that matches the numbers, and it looks like @Guillem_dlc can't either.
I read the 7kg m2 as being the MoI about the object's centre, which is not how it is interpreted in the OP. But that only makes the numbers crazier.
 
  • #4
haruspex said:
The problem description makes no sense to me.
Same here for at least 2 read-throughs. After a couple more, I'm thinking that the 2 masses are joined by a massless cable that goes over a pulley at a 90 degree angle and the pulley has an MOI that is involved in the problem. But that could be wrong assumptions on my part.

@Guillem_dlc -- Is this really how the whole problem was presented to you? There was no diagram, and you are supposed to come up with a sketch of your interpretation of the problem statement?
 
  • #5
berkeman said:
I'm thinking that the 2 masses are joined by a massless cable that goes over a pulley at a 90 degree angle
It says the cable is "rectilinear", i.e. straight.
 
  • #7
berkeman said:
Same here for at least 2 read-throughs. After a couple more, I'm thinking that the 2 masses are joined by a massless cable that goes over a pulley at a 90 degree angle and the pulley has an MOI that is involved in the problem. But that could be wrong assumptions on my part.

@Guillem_dlc -- Is this really how the whole problem was presented to you? There was no diagram, and you are supposed to come up with a sketch of your interpretation of the problem statement?
There was no diagram, no.
 
  • #8
Guillem_dlc said:
If we want ##\omega =6\, \textrm{rad}/\textrm{s}## and ##L=10\, \textrm{kg m}^2/\textrm{s}## ...
The system's total moment of inertia would be ##I = \frac L{\omega} = \frac {10}6 \approx 1.67 kg~m^2##.

This is smaller than the given MoI for the 1st object, which is impossible.
 
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  • #9
Guillem_dlc said:
There was no diagram, no.
Okay, then showing us your diagram is doubly-important. :wink:
 

FAQ: Two objects joined by a rectilinear cable rotating

1. What is the principle behind two objects joined by a rectilinear cable rotating?

The principle behind this scenario is based on the concept of rotational dynamics and tension in the cable. When two objects are connected by a rectilinear cable and one or both are rotated, the tension in the cable plays a crucial role in transmitting forces between the objects while maintaining their relative positions.

2. How does the length of the cable affect the rotation of the objects?

The length of the cable affects the radius of rotation of the objects. A longer cable allows for a larger radius, which can affect the angular velocity and the forces acting on each object. If the cable is too short, it may limit the range of motion and increase the tension, potentially leading to failure of the cable or the objects.

3. What factors influence the tension in the cable during rotation?

The tension in the cable is influenced by several factors, including the mass of the objects, the angular velocity of the rotation, the distance between the objects, and any external forces acting on them (such as gravity or friction). As the angular velocity increases, the tension in the cable typically increases as well, requiring careful consideration of the cable's material properties.

4. Can the objects rotate at different speeds when joined by a rectilinear cable?

No, the objects cannot rotate at different speeds while being joined by a rectilinear cable. The cable enforces a constraint that maintains a constant relationship between their angular velocities due to the tension in the cable. If one object rotates faster, it would create a difference in tension that would not be sustainable without affecting the rotation of the other object.

5. What are some practical applications of this system?

This system has several practical applications, including in mechanical systems such as pulleys, robotic arms, and amusement park rides. It can also be observed in certain engineering designs where objects need to maintain a specific orientation while rotating, such as in satellite systems or in certain types of machinery where synchronized rotation is necessary.

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