Motion of two constrained masses

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In summary, the conversation discusses a problem involving a spring and two objects, A and B. The initial conditions and equations for energy conservation and kinematic constraints are given. The goal is to find the minimum value of v_0 for which B hits the floor. The result is v_{0}\approx0.37 m/s, although there may be some discrepancy with the given right answer.
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PhMichael
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Homework Statement


The attached images shows everything. ([itex]{\bf{e}}_1[/itex] denotes the direction of X and [itex]{\bf{e}}_2[/itex] denotes the direction of Y).

Initially, the spring is force free when [itex]X_{0}=0.4 m[/itex] (which yields [itex]Y_{0}=0.3 m[/itex]). Also, at this instant, the velocity of B is [itex]{\bf{V}}_{B0}=-v_{0}{\bf{e}}_{2}[/itex] such that the corresponding velocity of A is [itex]{\bf{V}}_{A0}=v_{0}(Y/X){\bf{e}}_{1}=0.75v_0{\bf{e}}_{1}[/itex].

I'm asked to find the minimum value of [itex]v_0[/itex] for which B hits the floor.

Homework Equations


Energy conservation yields
[tex](\frac{1}{2}m_{B}V_B^{2}+\frac{1}{2}m_{A}V_A^{2})-(\frac{1}{2}m_{B}V_{B0}^{2}+\frac{1}{2}m_{A}V_{A0}^{2})+\frac{1}{2}k(X_{0}-L)^2=0[/tex]
However, the kinematic constrains (L is constant) yields
[tex]XV_{A}+YV_{B}=0[/tex]
So, when B hits the floor, [itex]Y=0[/itex] so that, at this instant, [itex]V_{A}=0[/itex]. Hence,
[tex]V_B^{2}=(V_{B0}^{2}+\frac{m_{A}}{m_{B}}V_{A0}^{2})-\frac{k}{m_B}(X_{0}-L)^2=v_{0}^{2}(1+\frac{2}{3}(0.75)^{2})-\frac{100}{3}(0.4-0.5)^2[/tex]
Now, my requirement is that [itex]V_B=0[/itex] when it hits the floor, so that the last equation yields
[tex]v_{0}\approx0.49 m/s[/tex]
while the right answer is
[tex]v_{0}\approx0.37 m/s[/tex]
Am I doing/assuming something wrong here?

I would appreciate your help!

The Attempt at a Solution


See the previous item.
 

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  • #2
While it would help if you show more steps, I agree with your result. 0.37 m/s would be the speed of A.
 

1. What is the definition of "motion of two constrained masses"?

The motion of two constrained masses refers to the movement of two objects that are connected or constrained in some way. This can include objects connected by a string, a rod, or any other type of physical constraint.

2. How does the motion of two constrained masses differ from the motion of a single object?

The motion of two constrained masses differs from the motion of a single object because the two masses are affected by each other's movements. This means that the motion of one mass will have an impact on the motion of the other mass, leading to a more complex and interdependent system.

3. What are some examples of systems that involve the motion of two constrained masses?

Some examples of systems that involve the motion of two constrained masses include a simple pendulum, a double pendulum, a pulley system, and a seesaw. These systems all involve two objects that are connected or constrained in some way and exhibit complex motion patterns.

4. What factors influence the motion of two constrained masses?

The motion of two constrained masses is influenced by various factors, including the masses of the objects, the length and stiffness of the connecting constraint, and any external forces acting on the system. These factors can impact the speed, direction, and overall behavior of the two masses.

5. How is the motion of two constrained masses mathematically described?

The motion of two constrained masses can be described using mathematical equations, such as Newton's laws of motion and the principles of conservation of energy and momentum. These equations help to predict the motion of the masses and understand how different factors affect their movement.

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