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- Homework Statement
- A flexible strip of length L is wound and its free end is fixed by a nail to the surface of an inclined plane that forms an angle θ with the horizontal. Then, the tape is allowed to unwind as it descends the slope.

How long will the tape be completely unrolled?

- Relevant Equations
- I = mR^2 /2

Hello everbody,

I found the following resolution:

We can consider the winding of the wire as a thin disc.

Like this,

For a given instant t, we will have M (t) mass of the disc and R (t) radius of the disc.

Analyzing linear dynamics:

$$M(t)gsen\theta - T = M(t).a $$

Analyzing the rotational dynamics:

Resulting torque = moment of inertia. angular acceleration (modularly)

Considering the center of the disk as the point of rotation, the only force that will make torque is traction.

$$T.R(t) = I \alpha = I . \frac{a}{R(t)}$$

The moment of inertia for a thin disc is given by:

$$I=\frac{M(t).R(t)^2}{2}$$

Replacing

$$T=\frac{M (t)R (t)^2.a}{2}$$

Substituting T, we find that the linear acceleration will be:

$$a=\frac{2}{3}.g.sen\theta$$

Realize that this linear acceleration is constant over time.

So, we can apply a cinematic:

$$S=\frac{at^2}{2}$$

Replacing S with L and isolating t:

$$t=\sqrt{\frac{3L}{gsin\theta} }$$

My question is:

Why in the analysis of the translation dynamics did you not consider the contribution of the system's mass variation? Since this is a variable mass system, shouldn't we use that in the tangential direction $$F = \frac{Mdv}{ dr} + \frac{udm}{dt}$$ where $$u$$ is the relative velocity of mass ejection in relation to the analyzed body?

I found the following resolution:

We can consider the winding of the wire as a thin disc.

Like this,

For a given instant t, we will have M (t) mass of the disc and R (t) radius of the disc.

Analyzing linear dynamics:

$$M(t)gsen\theta - T = M(t).a $$

Analyzing the rotational dynamics:

Resulting torque = moment of inertia. angular acceleration (modularly)

Considering the center of the disk as the point of rotation, the only force that will make torque is traction.

$$T.R(t) = I \alpha = I . \frac{a}{R(t)}$$

The moment of inertia for a thin disc is given by:

$$I=\frac{M(t).R(t)^2}{2}$$

Replacing

$$T=\frac{M (t)R (t)^2.a}{2}$$

Substituting T, we find that the linear acceleration will be:

$$a=\frac{2}{3}.g.sen\theta$$

Realize that this linear acceleration is constant over time.

So, we can apply a cinematic:

$$S=\frac{at^2}{2}$$

Replacing S with L and isolating t:

$$t=\sqrt{\frac{3L}{gsin\theta} }$$

My question is:

Why in the analysis of the translation dynamics did you not consider the contribution of the system's mass variation? Since this is a variable mass system, shouldn't we use that in the tangential direction $$F = \frac{Mdv}{ dr} + \frac{udm}{dt}$$ where $$u$$ is the relative velocity of mass ejection in relation to the analyzed body?