A completely filled barrel and its contents have a combined mass of 90 kg. A cylinder C is connected to the barrel at a height h = 550 mm as shown. Knowing μs = 0.40 and μk = 0.35, determine the maximum mass of C so the barrel will not tip.

Question

A completely filled barrel and its contents have a combined mass of 90 kg. A cylinder C is connected to the barrel at a height h= 550 mm as shown. Knowing [latex]{ \mu }_{ s }=0.40[/latex] and [latex]{ \mu }_{ k }=0.35[/latex], determine the maximum mass of C so the barrel will not tip.

Accepted Answer

Kinematics: Assume that the barrel is sliding, but not tipping.

[latex]\alpha=0\qquad\mathbf{a}_{G}=a\longrightarrow[/latex]

Since the cord is inextensible [latex]\quad{mathbf{a}}_{C}=a\downarrow[/latex]

Kinetics: Draw the free body diagrams of the barrel and the cylinder. Let T be the tension in the cord.

The barrel is sliding. [latex]\quad F_{F}=\mu_{k}N=0.35\,N[/latex]

Assume that tipping is impending, so that the line of action of the reaction on the bottom of the barrel passes through Point B.

[latex]e=250{\mathrm{~mm}}[/latex]

For the barrel.

[latex]\begin{aligned}& +\uparrow \Sigma F_y=0: N-W_{B}=0\quad\quad N=W_{B}=m_{B}g=882.90~\mathrm{N}\& +\circlearrowleft \Sigma M_{G}=0;\quad N e-100T-(450)(0.35\,N)=0\& \quad\quad\quad\quad T={\frac{e-(450)(0.35)}{100}}N={\frac{250-157.5}{100}}(882.90)=816.68\mathrm{{\ N}}\& \overset{+}{\longrightarrow } \Sigma F_{x}=m_{B}a;\quad T-0.35\,N=m_{B}a\& \quad\quad\quad\quad a=\frac{8\,16.68}{90}-0.35\frac{882.90}{90}=5.6407\ \mathrm{m/s}^{2}\end{aligned}[/latex]

For the cylinder:

[latex]\begin{aligned}& +\downarrow\Sigma F=m_{C}a\colon\;\;W_{C}-T=m_{C}a\& \quad\quad\quad\quad m_{C}g-T=m_{C}a\& \quad\quad\quad\quad\quad\quad\quad\quad m_{C}=\frac{T}{g-a}=\frac{816.68}{9.81-5.6407}=195.88\ \mathrm{kg}\&& m_{C}=195.9\,\mathrm{kg}\blacktriangleleft\end{aligned}[/latex]

Question 16.11: A completely filled barrel and its contents have a combined ...

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