A tractor pulls a harrow (a ploughing tool) of mass m₁ with a net force F resulting in an acceleration of a₁. The same tractor pulls a trolley of mass m₂ with a force F producing an acceleration of a₂. If the tractor now pulls the trolley with the harrow placed on it (with the same force F), then obtain an expression for the resulting acceleration in terms of a₁ and a₂. Ignore friction.

Given, a tractor pulls a harrow of mass $m_1$ with force $F$ producing acceleration $a_1$, and a trolley of mass $m_2$ with the same force $F$ producing acceleration $a_2$.

Using Newton's second law,

For the harrow,

$$F = m_1 a_1$$

$$m_1 = \frac{F}{a_1}$$

For the trolley,

$$F = m_2 a_2$$

$$m_2 = \frac{F}{a_2}$$

When the tractor pulls both together, total mass,

$$m = m_1 + m_2$$

$$m = \frac{F}{a_1} + \frac{F}{a_2}$$

If the resulting acceleration is $a$, then

$$F = ma$$

$$F = \left( \frac{F}{a_1} + \frac{F}{a_2} \right) a$$

$$1 = \left( \frac{1}{a_1} + \frac{1}{a_2} \right) a \Rightarrow a = \frac{a_1 a_2}{a_1 + a_2}$$

Therefore, the resulting acceleration is

$$a = \frac{a_1 a_2}{a_1 + a_2}$$