Disturbed Atwood machine

Problem

An Atwood machine has $m_1 = 3\ \text{kg}$ and $m_2 = 5\ \text{kg}$, an ideal pulley and an ideal string. At a certain instant an ant of mass $\delta m = 0{,}01\ \text{kg}$ crawls from $m_1$ to $m_2$. Calculate the percentage change in the system’s acceleration during the crossing (assume the ant stays on $m_2$).

Solution

For an ideal Atwood machine, applying Newton’s second law to both masses and recalling that the string tension is equal at both ends, the system’s acceleration is: $a = \frac{(m_2 - m_1)\,g}{m_1 + m_2}$

The total mass hung from the pulley ($m_1 + m_2 + \delta m = 8{,}01\ \text{kg}$) stays unchanged during the ant’s crossing: what changes is on which side the small mass $\delta m$ sits.

Before the crossing (ant on $m_1$): the “light” side weighs $m_1 + \delta m = 3{,}01\ \text{kg}$, the “heavy” side $m_2 = 5\ \text{kg}$. The difference is $5 - 3{,}01 = 1{,}99\ \text{kg}$: $a_i = \frac{(5 - 3{,}01)\,g}{8{,}01} = \frac{1{,}99 \cdot 9{,}81}{8{,}01} \approx 2{,}437\ \text{m/s}^2$

After the crossing (ant on $m_2$): the light side weighs $m_1 = 3\ \text{kg}$, the heavy side $m_2 + \delta m = 5{,}01\ \text{kg}$. The difference rises to $5{,}01 - 3 = 2{,}01\ \text{kg}$: $a_f = \frac{(5{,}01 - 3)\,g}{8{,}01} = \frac{2{,}01 \cdot 9{,}81}{8{,}01} \approx 2{,}462\ \text{m/s}^2$

Percentage change. The difference in acceleration is: $\Delta a = a_f - a_i = 2{,}462 - 2{,}437 = 0{,}0245\ \text{m/s}^2$

Relative to the initial value: $\frac{\Delta a}{a_i} = \frac{0{,}0245}{2{,}437} \approx 0{,}010$

$\ev{\frac{\Delta a}{a} \approx 1\%}$

The effect is surprisingly large: shifting just $10\ \text{g}$ between the two sides changes the acceleration by $1\%$. The reason is that what matters is the difference $m_2 - m_1 = 2\ \text{kg}$, small compared with the masses involved; a tiny ant switching sides changes this difference by $2\,\delta m = 0{,}02\ \text{kg}$, i.e. about $1\%$. The system is so sensitive because it operates close to the balance point between the two masses.

Links

Topics: Dynamics Concepts: Newton’s second law Skills: Analysis of limiting cases and science-fiction physics Objects: Atwood machine