Relating displacements in a pulley system

In summary, we can use energy conservation to relate the two displacements, $dx_1$ and $dx_2$, as shown in the equation $Tdx_1= 2Tdx_2 - Mg dx_2$. This simplifies to $dx_1 = (2- \frac{Mg}{T} ) dx_2$ and can also be derived using the differential conservation of rope.
  • #1
burian
64
6
> The set up: At left end, the rope is pulled down with a distance $dx_1$ by a force of constant magnitude $F$, the mass of $M$ is wrapping around by rope on the right and moves up by a distance $dx_2$ due to this. Problem: Find relate the two displacements.

I thought of applying energy conservation, we put in energy $Tdx_1$ into the system and we add up the energy induced on the rest of the system. Noting that $F=T$,

$$Tdx_1= 2Tdx_2 - Mg dx_2$$

**Explanation for left side** : We input an energy of $F \cdot dx_1$ when we pull the rope by$dx_1$ with force $F$, since $F=T$, the energy is just $T dx_1$

**Explanation for right side :** The mass is pulled up by $dx_2$, this goes into kinetic energy of the body, this kinetic energy can be written using the work energy theorem as the external forces dotted with $dx_2$, $(2T-mg) \hat{j} \cdot (dx_2 \hat{j})= (2T- mg) dx_2$

This simplifies to:

$$ dx_1 = (2- \frac{Mg}{T} ) dx_2 \tag{1}$$
But, if we go by the differential conservation of rope, we find that $dx_1 = 2 dx_2$ is it possible to simplfy eqtn (1) into this, or have I done something conceptually wrong?
 

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  • #2
Not sure why inline latex is not working..
 
  • #3
You need to use double $ sign.
$$F=T$$
 
  • #4
bump
 

Related to Relating displacements in a pulley system

1. How do you calculate the displacement in a pulley system?

The displacement in a pulley system can be calculated by using the formula: displacement = (number of pulleys)(change in length of rope per pulley). This formula assumes that the rope does not stretch or slip.

2. What is the relationship between the displacement of the load and the displacement of the effort in a pulley system?

In a pulley system, the displacement of the load is equal to the displacement of the effort. This is due to the fact that the rope is continuous and inelastic, meaning that any movement of one end of the rope will result in an equal movement of the other end.

3. Can the direction of displacement in a pulley system be changed?

Yes, the direction of displacement in a pulley system can be changed by changing the direction of the effort. This can be achieved by changing the direction of the force applied to the rope or by changing the direction of the pulley itself.

4. How does the number of pulleys in a system affect the displacement?

The number of pulleys in a system does not affect the displacement of the load or the effort. However, it does affect the amount of effort required to move the load. The more pulleys there are, the less effort is required to lift the load.

5. What is the difference between single and multiple pulley systems in terms of displacement?

In a single pulley system, the displacement of the load is equal to the displacement of the effort. However, in a multiple pulley system, the displacement of the load is equal to the displacement of the effort divided by the number of pulleys. This means that in a multiple pulley system, the load will move a shorter distance compared to the effort.

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