# Calculating Tensions and Acceleration in Frictionless Pulley System

• Lamoid
In summary, the conversation discusses the problem of finding the tensions and acceleration of two bodies connected by a string and pulley system. The diagram provided shows that the blocks are moving in opposite directions, with the acceleration of block B being half of block A. The reason for this can be proven mathematically by dividing the string into three pieces and using the fact that the string length remains constant. Alternatively, one can experiment with a piece of string to see the effect of folding it in half.
Lamoid
The weights of the objects are 200 N and 300 N. The pulleys are essentially frictionless and massless. P1 has a stationary axle but P2 is free to move up and down. Find the tensions FT1 and FT2 and the acceleration of each body.

Diagram I made in paint attached.

Now my problem is not that I can't get the right answer but that I don't understand why. It gives me a hint that the acceleration of block B is half (and in the opposite direction of course) that of block A and I cannot not figure out why that is. Can anyone explain it to me?

And yes, the diagram sucks but the A block is moving down, B is moving up.

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You can prove mathematically that the magnitude of the acceleration of A is twice that of B using the fact that the string length remains constant. Divide the string into three pieces: X (from mass A to P1), Y (from P1 to P2), Y (from P2 to the spot of rope at the height of P1). Thus the length of the rope is X + 2Y. Since the length can't change, take the derivative twice to find the acceleration constraint: a_x + 2a_y = 0. Make sense? (a_x is the acceleration of mass A; a_y is the acceleration of mass B)

But even better than proving it is to just play around with a piece of string and see how folding the string in half affects things.

Thank you very much! That's a neat calculus proof.

## 1. How do you determine the tension in a frictionless pulley system?

To calculate the tension in a frictionless pulley system, you need to first identify all the forces acting on the system. Then, use Newton's second law (F=ma) to determine the acceleration of the system. Finally, use the relationship T=ma, where T is the tension and a is the acceleration, to find the tension in the system.

## 2. What is the role of friction in a pulley system?

Friction plays no role in a frictionless pulley system. This means that there is no resistance or force opposing the motion of the pulleys or the objects being pulled by the system. This allows for a simplified analysis of the system, making it easier to calculate the tensions and accelerations.

## 3. Can the tension in a frictionless pulley system be negative?

No, the tension in a frictionless pulley system cannot be negative. Tension is a force and forces cannot be negative. However, the direction of the tension force can be negative if it is acting in the opposite direction of the system's motion.

## 4. How do you handle multiple pulleys in a frictionless pulley system?

To handle multiple pulleys in a frictionless pulley system, you can use the concept of mechanical advantage. The mechanical advantage is the ratio of the output force (tension) to the input force (applied force). In a frictionless system, the mechanical advantage is equal to the number of ropes supporting the load. This means that for every additional pulley, the tension will decrease by a factor of two.

## 5. Can the acceleration in a frictionless pulley system be zero?

Yes, the acceleration in a frictionless pulley system can be zero if all the forces acting on the system are balanced. This means that the net force is equal to zero, resulting in zero acceleration. In this case, the tension in the system will also be zero as there is no need for a force to pull the objects in the system.

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