# Constrained Motion of 7 Masses and 3 Pulleys

• Engineering
• mingyz0403
In summary, when setting the directions for this problem, it does not matter which direction is chosen as positive as long as consistency is maintained. The error in the answer was due to assigning a negative value to a variable, and it is better to assume all variables as positive and let the result determine the direction. The given data shows that block A's acceleration is actually in the opposite direction as initially assumed.
mingyz0403
Homework Statement
Block B starts from rest, block A moves with a constant acceleration, and slider block C moves to the right with a constant acceleration of 75 mm/s2. Knowing that at t = 2 s the velocities of B and C are 480 mm/s downward and 280 mm/s to the right, respectively, use kinematics to determine:
a. The acceleration of A and B
b. The initial velocities of A and C
c. The change in positions of slider C after 3 s.
Relevant Equations
Motion equation.
Conservation of string
How do you set the direction for this problem? Do you look at the motion of the block? If you set right positive, does that automatically makes upward motion positive? I set right as positive and up as positive for this problem. However, my answer was wrong. Textbook solution set right as positive and down as positive. I don't understand why setting right as positive and up as positive wrong for this problem.

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mingyz0403 said:
... I don't understand why setting right as positive and up as positive wrong for this problem.
I don’t believe that setting the directions of positive one way or the other makes a difference in the result.
The sign of the value that you obtain at the end always tells you whether or not your initial assumption regarding the direction of that magnitude was correct.

Your error was assigning a negative value to ##4a_b##.
When in doubt, it is better to assume all as positive and let the result decide.

The original data is telling us that:
1) The length Xc is growing as block C is moving to the right.
2) The four lengths Xb are growing as block B is moving downwards.
3) The three lengths Xa are shrinking as block A can only move upwards as result of conditions 1) and 2) above.
\begin{align} a_C+4a_B+3a_A&=0\nonumber\\ 75+4(480/2)+3a_A&=0\nonumber\\ -75-960&=3a_A\nonumber\\ -1035/3&=a_A\nonumber\\ a_A&=-345~mm/s^2\nonumber\\ \end{align}
Note that the negative sign here means that the actual direction of the acceleration of block A is opposed to the originally assumed.

## 1. What is "Constrained Motion of 7 Masses and 3 Pulleys"?

The Constrained Motion of 7 Masses and 3 Pulleys is a physics problem that involves a system of 7 masses connected by strings over 3 pulleys. The motion of the masses is constrained by the strings and pulleys, making it a complex and challenging problem to solve.

## 2. What are the key principles involved in solving this problem?

The key principles involved in solving the Constrained Motion of 7 Masses and 3 Pulleys problem are Newton's laws of motion, conservation of energy, and the equations of motion for systems with multiple masses and pulleys. These principles provide the foundation for understanding the forces and motion of the masses in the system.

## 3. How do you approach solving this problem?

Solving the Constrained Motion of 7 Masses and 3 Pulleys problem involves breaking down the system into smaller, more manageable parts. This can be done by drawing free-body diagrams for each mass and analyzing the forces acting on them. Then, using the principles mentioned above, equations can be set up and solved to determine the motion of the masses.

## 4. What are some common challenges in solving this problem?

One of the main challenges in solving the Constrained Motion of 7 Masses and 3 Pulleys problem is keeping track of all the forces and their directions in the system. It can also be difficult to determine the correct equations to use for each mass, as well as properly setting up the initial conditions for the system.

## 5. What are some real-world applications of this problem?

The Constrained Motion of 7 Masses and 3 Pulleys problem has many real-world applications, such as in engineering and robotics. It can be used to model the motion of complex systems involving multiple masses and pulleys, such as cranes, elevators, and conveyor belts. Solving this problem can also help in understanding the dynamics of more advanced machines and systems.

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