Two disks sliding on a frictionless horizontal plane with opposite velocities of the same magnitude ν0 hit each other squarely. Disk A is known to have a weight of 6-lb and is observed to have zero velocity after impact. Determine (a) the weight of disk B, knowing that the coefficient of

Question

Two disks sliding on a frictionless horizontal plane with opposite velocities of the same magnitude [latex] u_{ 0 }[/latex] hit each other squarely. Disk A is known to have a weight of 6-lb and is observed to have zero velocity after impact. Determine (a) the weight of disk B, knowing that the coefficient of restitution between the two disks is 0.5, (b) the range of possible values of the weight of disk B if the coefficient of restitution between the two disks is unknown.

Accepted Answer

Total momentum conserved:

[latex]\begin{aligned}\stackrel{+}{\longrightarrow} m_{A} u_{A}+m_{B} u_{B} & =m_{A} u_{A}^{\prime}+m_{B} u^{\prime} \\left(m_{A} ight) u_{0}+m_{B}\left(- u_{0} ight) & =0+m_{B} u^{\prime} \ u^{\prime} & =\left\lgroup\frac{m_{A}}{m_{B}}-1 ight group u_{0}\quad\quad ext{(1)}\end{aligned}[/latex]

Relative velocities:

[latex]\begin{aligned} u_{B}^{\prime}- u_{A}^{\prime} & =e\left( u_{A}- u_{B} ight) \ u^{\prime} & =2 e u_{0}\quad\quad ext{(2)}\end{aligned}[/latex]

Subtracting Eq. (2) from Eq. (1) and dividing by [latex] u_{0}[/latex],

[latex]\frac{m_{A}}{m_{B}}-1-2 e=0 \quad \frac{m_{A}}{m_{B}}=1+2 e \quad m_{B}=\frac{m_{A}}{1+2 e}[/latex]

Since weight is proportional to mass, [latex]\quad\quad \quad W_{B}=\frac{W_{A}}{1+2e}[/latex](3)

(a) With [latex]W_{A}=6 \mathrm{\ lb}[/latex] and e = 0.5,

[latex]W_{B}=\frac{6}{1+(2)(0.5)}\quad\quad\quad W_{B}=3.00 \mathrm{\ lb}\blacktriangleleft[/latex]

(b) With [latex]W_{A}=6 \mathrm{\ lb}[/latex] and e = 1,

[latex]W_{B}=\frac{6}{1+(2)(1)}=2 \mathrm{\ lb}[/latex]

With [latex]W_{A}=6 \mathrm{\ lb}[/latex] and e = 0,

[latex]W_{B}=\frac{6}{1+(2)(0)}=6 \mathrm{\ lb}[/latex]

Range: [latex]\quad\quad \quad\quad 2.00 \mathrm{\ lb} \leq W_{B} \leq 6.00 \mathrm{\ lb}\blacktriangleleft[/latex]

Question 13.158: Two disks sliding on a frictionless horizontal plane with op...

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