- #1
AbigailM
- 46
- 0
Homework Statement
Two blocks A and B with respective masses [itex]m_{A}[/itex] and [itex]m_{B}[/itex] with respective masses [itex]m_{A}[/itex] and [itex]m_{B}[/itex] are connected via a string. Block B is on a frictionless table, and block A is hanging at a vertical distance h from a spring with spring constant k that is at its equilibrium position. The blocks are initially at rest. Find the velocity of A and B when the spring is compressed by an amount [itex]\delta y =m_{A}g/k[/itex]. Determine the maximum compression [itex]\delta y_{max} of the spring in terms of [itex]m_{A}, m_{B}[/itex], g and k. (Hint: what happens to the motion of the blocks when [itex]\delta y-m_{A}g/k[/itex]?)
Homework Equations
[itex]\delta y=m_{A}g/k[/itex] (Eq 1)
[itex](m_{A}+m_{B})gh=\frac{1}{2}(m_{A}+m_{B})v^{2}[/itex] (Eq2)
[itex]\frac{1}{2}m_{A}v^{2}=m_{A}g\delta y - \frac{1}{2}k\delta y^{2}[/itex] (Eq 3)
The Attempt at a Solution
From Eq2 [itex]v_{B}=\sqrt{2gh}[/itex]
Solve Eq3 for v and substitute in Eq1. Then we can subtract our new equation from Eq2:
[itex]v_{A}=\sqrt{2gh}-\sqrt{m_{A}/k}g[/itex]
To find [itex]\delta y_{max}[/itex] substitute [itex]v=\sqrt{2gh}[/itex] into Eq3:
[itex]m_{A}gh=m_{A}g\delta y - \frac{1}{2}k\delta y^{2}[/itex]
Now solve for [itex]\delta y[/itex]:
[itex]\delta y=\frac{m_{A}g-\sqrt{m_{A}^{2}g^{2}-2km_{A}g}}{k}[/itex]
Does this look correct? Thanks for the help