Day: May 23, 2023

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202305231027 Problem 8.21-22

In the figure below, two objects are connected by a string which is threaded through a pulley.

Using its weight, object 2 moves object 1 along a flat surface. The acceleration a of the two objects can be determined by the following formula

\displaystyle{a=\frac{m_2g-\mu m_1g}{m_1+m_2}}

where m_1 and m_2 are the masses of object 1 and object 2, respectively, in kilograms, g is the acceleration due to Earth’s gravity measured in \displaystyle{\frac{\mathrm{m}}{\mathrm{sec}^2}}, and \mu is a constant known as the coefficient of friction. Which of the following expresses \mu in terms of the other variables?

A. \displaystyle{\mu=\frac{a(m_1+m_2)}{m_1m_2g^2}}
B. \displaystyle{\mu=\frac{a(m_1+m_2)}{m_2g-m_1g}}
C. \displaystyle{\mu=\frac{m_2g-a(m_1+m_2)}{m_1g}}
D. \displaystyle{\mu=\frac{a(m_1+m_2)-m_2g}{m_1g}}

If the masses of both object 1 and object 2 were doubled, how would the acceleration of the two objects be affected?

A. The acceleration would stay the same.
B. The acceleration would be halved.
C. The acceleration would be doubled.
D. The acceleration would be quadrupled (multipled by a factor of 4).

Extracted from Phu Nielson. (2015). SAT Math Advanced Guide and Workbook.

Roughwork.

First, begin with free-body diagrams:

Next, carry on by Newton’s 2nd law:

\mathbf{F}_{\mathrm{net}}=m\mathbf{a}

jotting positive the motion directing to,

\begin{aligned} (T-\mu m_1g)\,\hat{\mathbf{i}} & = m_1a\,\hat{\mathbf{i}} \\ (m_2g-T)\,\hat{\mathbf{j}} & = m_2a\,\hat{\mathbf{j}} \\ \end{aligned}

then writing Euler-Lagrange (E-L) equation with Rayleigh dissipation function \mathcal{F}:

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t}\bigg(\frac{\partial\mathcal{L}}{\partial\dot{x}}\bigg) -\frac{\partial\mathcal{L}}{\partial x} = -\frac{\partial\mathcal{F}}{\partial\dot{x}}}

where

\begin{aligned} \mathcal{L} & = T-V \\ T & = \frac{1}{2}(m_1+m_2)\dot{x}^2 \\ V & = m_2gx \\ \mathcal{F} & = k\dot{x}^2 \enspace (\leqslant \mu m_1g) \\ \end{aligned}

or by noting the undissipated Hamiltonian \mathcal{H}(\mathbf{p},\mathbf{q})=T+V as in:

\begin{aligned} \frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t} & = \frac{\partial\mathcal{H}}{\partial\mathbf{p}} \\ \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} & = -\frac{\partial\mathcal{H}}{\partial\mathbf{q}} \\ \end{aligned}

where though it is not uncommon for some few to despise differentiating with respect to vectors whom I hereby ignore.

\displaystyle{\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+m_2gq-\mu m_1gq}.

This problem is not to be attempted.