Double Differentiation and Acceleration in a Four Pulley System

In summary, the conversation discusses the acceleration of block B, which can be obtained by differentiating the y displacement twice with respect to time t. The resulting acceleration is 1/2 m/s^2. However, the position of B in relation to A is not as simple, as demonstrated by a series of pulleys and their corresponding movements. This leads to a discussion on faulty logic and the need for experimentation or visual aids to understand the concept better.
  • #1
rudransh verma
Gold Member
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Homework Statement
The vertical displacement of block A in meters is ##y=\frac{t^2}4## where t is in sec. Calculate the downward acceleration of block B.
Relevant Equations
##F=ma##
We can differentiate twice the y displacement with respect to time t and get the acceleration of block B. $$a_B= \frac12 m/s^2$$.
But I don’t think it’s that simple.
 

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  • #2
You think correctly :smile:
But at least you have the acceleration of A.
How is the position of B in relation to A ? (if A moves up 1 m, how much does B move?)

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  • #3
BvU said:
You think correctly :smile:
:smile:
BvU said:
How is the position of B in relation to A ? (if A moves up 1 m, how much does B move?)
Of course 1 m since all are connected with unstrechable ropes.
 
  • #4
Your logic is faulty. Think again. Number the pulleys 1,2,3,4 left to right. If 1 goes up by 1 unit, 2 stays in position unchanged, 3 ... etc.

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  • #5
BvU said:
Your logic is faulty. Think again. Number the pulleys 1,2,3,4 left to right. If 1 goes up by 1 unit, 2 stays in position unchanged, 3 ... etc.

##\ ##
Oh like that!
1 goes up by 1 unit then 2 stays in place, 3 and 4 goes down by 1 unit.
 
  • #6
2 correct
3 think again, perhaps do the experiment..., or make 2 drawings...

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1. What is double differentiation in a four pulley system?

Double differentiation refers to the process of taking two successive derivatives of a function in a four pulley system. This allows for the calculation of the acceleration of the system, which can be useful for analyzing the motion of objects and determining the forces acting on them.

2. How is acceleration calculated in a four pulley system?

Acceleration can be calculated in a four pulley system by taking the second derivative of the position function with respect to time. This involves taking the first derivative to find the velocity, and then taking the derivative of the velocity to find the acceleration.

3. What is the significance of a four pulley system in double differentiation?

A four pulley system is significant in double differentiation because it allows for the calculation of acceleration in complex systems where multiple pulleys are involved. This can be useful in engineering and physics applications, such as analyzing the motion of a pulley-driven machine.

4. What are the limitations of using double differentiation in a four pulley system?

There are a few limitations to using double differentiation in a four pulley system. One limitation is that it assumes the system is in equilibrium, meaning that the forces acting on the pulleys are balanced. It also assumes that the pulleys are massless and there is no friction present.

5. How can double differentiation in a four pulley system be applied in real-world situations?

Double differentiation in a four pulley system can be applied in real-world situations to analyze the motion of objects and determine the forces acting on them. This can be useful in fields such as engineering, physics, and mechanics, where understanding the acceleration of a system is important for designing and optimizing machines and structures.

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