Calculating Time for Window Shade Roll-Up: M, W, L, T, R, m, g Dynamics

In summary, the conversation discusses a complex problem involving a window shade of certain mass, thickness, and length being driven by a linear spring. The shade is rolled up on a solid cylinder with a specific radius and mass. The question asks for the time it takes for the shade to roll up completely from its unfurled position, assuming no friction or gravitational acceleration. The solution involves determining the radius and length of the rolling shade, calculating the moment of inertia, and solving a nonlinear equation using Newton's 2nd law. This problem is difficult to solve without a computer and further details and equations are available upon request.
  • #1
Loren Booda
3,125
4
A window shade of mass M, thickness W and length L is driven by a linear spring having tension T (with shade unfurled) and tension 0 (with shade rolled up). The solid cylinder upon which the shade winds has radius R and mass m. What is the time elapsed for the shade to roll up completely when released from its unfurled position? Assume zero friction and gravitational acceleration g.
 
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  • #2
To the best of my knowledge, this gets real ugly. The equations are easy to obtain, but because of their nonlinearity you'll have a hard time solving them. I think a numerical solution would more appropriate for this problem. I can however point you in the right direction.

First, you need to determine how the shade rolls around the cylinder: radius of the (cylinder + shade) system while it's rolling and length around the cylinder for any angle. The length is obtained by integrating the radius function.

Next, calculate the (angle dependant) moment of inertia. Fairly easy once you know the thickness of the shade on the cylinder.

Finally, find the expressions for the forces acting on the cylinder (weight of remaining shade and spring). Don't forget they depend on the angle. Plug them in Newton's 2nd law:
[tex]
I\left( \theta \right)\ddot \theta = \sum {\tau \left( \theta \right)}
[/tex]
You now need to solve the resulting equation. Not easy.

That's about it. If you need some more details (like equations), let me know :smile:
 
  • #3
Thanks, mu. I got my inspiration while tugging the shade of my bathroom window. Such a mundane process, fairly easily stated, but hard as heck to solve - one needs a computer. I am aware of the basic equations, just stumped on how they are ultimately coupled (nonlinearly).
 

1. How do you calculate the time it takes for a window shade to roll up?

The time it takes for a window shade to roll up can be calculated using the formula: T = (L + W) / (M * R * m * g). Where T is the time in seconds, L is the length of the shade, W is the width of the shade, M is the mass of the shade, R is the radius of the shade roll, m is the coefficient of friction, and g is the acceleration due to gravity.

2. What is the purpose of the variables M, W, L, T, R, m, and g in the equation?

M, W, and L represent the physical characteristics of the window shade, including its size and weight. T is the time it takes for the shade to roll up. R is the radius of the shade roll, which affects the amount of rotation needed to roll up the shade. m is the coefficient of friction, which determines how easily the shade can roll up. g is the acceleration due to gravity, which affects the weight of the shade and its movement.

3. How does the length and width of the window shade affect the roll-up time?

The length and width of the window shade are directly proportional to the roll-up time. This means that as the size of the shade increases, the roll-up time also increases. This is because a larger shade will have more mass and require more rotation to roll up, increasing the time it takes.

4. Does the mass of the shade affect the roll-up time?

Yes, the mass of the shade does affect the roll-up time. A heavier shade will require more force to roll up, increasing the time it takes. However, the effect of mass on the roll-up time is dependent on the other variables in the equation, such as the radius of the shade roll and the coefficient of friction.

5. How can the roll-up time be reduced?

The roll-up time can be reduced by adjusting the variables in the equation. This can include using a smaller radius for the shade roll, reducing the coefficient of friction by lubricating the shade mechanism, or reducing the mass of the shade. Additionally, using a motorized shade mechanism can also significantly decrease the roll-up time.

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