Dynamics -- Velocity of a Block in a system of Pulleys

In summary: We can then write this as: $$\frac{d}{dt}(\delta_D-\delta_A)=-{\frac {d}{dt}(\delta_B-\delta_A)}$$. Now, since the string will stay constant in length, $${\frac {d}{dt}(\delta_B-\delta_A)}=0$$. Therefore, we have: $$\frac{d}{dt}(\delta_B-\delta_A)=\frac{d}{dt}(\delta_C-\delta_A)+\frac{d}{dt}(\delta_D-\delta_
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Hey guys, I have been a long time lurker here but this problem has eluded me for a few hours now and my frustration at not being able to figure out what it is I am missing prompted me to finally pull the trigger on making an account here. I hope the way I wrote everything out is clear to understand but if it is not I would be happy to do my best to elaborate further. Thank you!

1. Homework Statement

The system shown starts from rest and each component moves with a constant acceleration. The relative acceleration of block C with respect to collar B is 60mm/s^2 upward and the relative acceleration of block D with respect to block A is 110mm/s^2 downward. Determine the velocity of block C after 6s.
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2. Homework Equations
Relative Acceleration: ac = ac/b + ab
v = v0 + a*t


The Attempt at a Solution


My first thought to solve this problem involved deriving an equation for the length of the cable that connects blocks A B and C that I could differentiate twice to get an acceleration equation and an equation for the length of the cable that connects blocks A B and D to do the same with. For the length of cable 1 I got 2SA + 2SB + SC = Constant taking my reference point to be the line that goes through the center of the 3 top pulleys. For the other cable I wasn't 100% certain what it should be though for what the question is asking I don't believe it is necessary to know. I can't seem to find any way to utilize this equation for the length of the cable to give me an acceleration for block B that I can use combined with the relative velocity equation to give me the acceleration for block C that I can use with the kinematic equation to find the velocity. My next thoughts for this problem was to find some way to use the given relative acceleration to simply obtain accelerations for the blocks directly however if this is possible then despite my best efforts I haven't found a way to do so. Any help at solving this would be GREATLY appreciated. I feel like it is easier than I am making it out to be which makes it all the more frustrating that I cannot solve it.[/B]
 
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  • #2
Let ##\delta_A##, ##\delta_B##, and ##\delta_D## be the downward displacements of A, B, and D respectively. In order for the bottom string to remain constant in length, $$(\delta_D-\delta_A)+(\delta_D-\delta_B)=0$$
 

FAQ: Dynamics -- Velocity of a Block in a system of Pulleys

What is the formula for calculating the velocity of a block in a system of pulleys?

The formula for calculating the velocity of a block in a system of pulleys is v = (F/g) x (2n/(n+1)), where v is the velocity, F is the applied force, g is the acceleration due to gravity, and n is the number of pulleys in the system.

How does the number of pulleys affect the velocity of the block in a system of pulleys?

The number of pulleys in a system directly affects the velocity of the block. As the number of pulleys increases, the velocity of the block decreases. This is because more pulleys means more rope to pull through, resulting in a larger distance traveled by the block for the same amount of force applied.

Can the velocity of the block in a system of pulleys ever be greater than the applied force?

No, the velocity of the block in a system of pulleys can never be greater than the applied force. This is because some of the applied force is always lost due to friction and the weight of the pulleys.

What factors can affect the velocity of a block in a system of pulleys?

The velocity of a block in a system of pulleys can be affected by the applied force, the number of pulleys, the weight and friction of the pulleys, and the weight of the block itself.

Can the velocity of the block in a system of pulleys be negative?

Yes, the velocity of the block in a system of pulleys can be negative. This would occur if the direction of the applied force is opposite to the direction of motion of the block. In this case, the block would be moving in the opposite direction of the force and have a negative velocity.

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