The motor at C pulls in the cable with an acceleration ac= (3 t^2) m/s^2, where t is in seconds. The motor at D draws in its cable at aD = 5 m/s^2. If both motors start at the same instant from rest when d = 3 m , determine (a) the time needed for d = 0 , and (b) the relative velocity of block A

Question

The motor at C pulls in the cable with an acceleration a_{C}=\left(3 t^{2}\right) \mathrm{m} / \mathrm{s}^{2} , where t is in seconds. The motor at D draws in its cable at a_{D} = 5 m/{s}^{2} . If both motors start at the same instant from rest when d = 3 m , determine (a) the time needed for d = 0 , and (b) the relative velocity of block A with respect to block B when this occurs.

Accepted Answer

for A :

[latex]s_{A}[/latex]+[latex]\left(s_{A}-s_{C}\right)[/latex]=l$
2 [latex]v_{A}[/latex]=[latex]v_{c}[/latex]
2 [latex]a_{A}[/latex]=[latex]a_{C}[/latex]=-3 [latex]t^{2}[/latex]
[latex]a_{A}[/latex]=-1.5 [latex]t^{2}[/latex]=1.5 [latex]t^{2}[/latex] [latex]\rightarrow[/latex]
[latex]v_{A}[/latex]=0.5 [latex]t^{3}[/latex] [latex]\rightarrow[/latex]
[latex]s_{A}[/latex]=0.125 [latex]t^{4}[/latex] [latex]\rightarrow[/latex]
For B:
[latex]a_{B}[/latex]=5 m/[latex]{s}^{2}[/latex]
[latex]v_{B}[/latex]=5 t
[latex]s_{B}[/latex]=2.5 [latex]t^{2}[/latex]
Require [latex]s_{A}[/latex]+[latex]s_{B}[/latex]=d
0.125 [latex]t^{4}[/latex]+2.5 [latex]t^{2}[/latex]=3
Set u=[latex]t^{2}[/latex] 0.125 [latex]u^{2}[/latex]+2.5 u=3
The positive root is u=1.1355 . Thus,
t=1.0656=1.07 s
[latex]v_{A}[/latex]=0.5[latex](1.0656)^{3}[/latex]=0.6050
[latex]v_{B}[/latex]=5(1.0656)=5.3281 m/s
[latex]{v}_{A}[/latex]=[latex]{v}_{B}[/latex]+[latex]{v}_{A / B}[/latex]
0.6050 i=-5.3281 i+[latex]v_{A / B}[/latex]i
[latex]v_{A / B}[/latex]=5.93 m/s [latex]\rightarrow[/latex]

Question : The motor at C pulls in the cable with an acceleration ac= (...

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