Question

A constant force F acts on the mass m₁and m₂ to produce respective acceleration of a₁ and a₂ . If same force acts on the object of mass ( m₁+m₂), then prove that the magnitude of acceleration produced is (a₁a₂)/a₁+a₂

Answer 1

$$\begin{gathered} \mathrm{Acceleration}\mathrm{of}\mathrm{first}\mathrm{mass}: \\ {\mathrm{a}}_1=\frac{\mathrm{F}}{{\mathrm{m}}_1} \\ \mathrm{Then},{\mathrm{m}}_1=\frac{\mathrm{F}}{{\mathrm{a}}_1} \\ \mathrm{Acceleration}\mathrm{of}\mathrm{second}\mathrm{mass}: \\ {\mathrm{a}}_2=\frac{\mathrm{F}}{{\mathrm{m}}_2} \\ \mathrm{Then},{\mathrm{m}}_2=\frac{\mathrm{F}}{{\mathrm{a}}_2} \\ \mathrm{Acceleration}\mathrm{of}\mathrm{combined}\mathrm{mass}: \\ \mathrm{a}=\frac{\mathrm{F}}{{\mathrm{m}}_1+{\mathrm{m}}_2} \\ =\frac{\mathrm{F}}{\frac{\mathrm{F}}{{\mathrm{a}}_1}+\frac{\mathrm{F}}{{\mathrm{a}}_2}} \\ =\frac{\mathrm{F}}{\mathrm{F}\left[\frac{1}{{\mathrm{a}}_1}+\frac{1}{{\mathrm{a}}_2}\right]} \\ =\frac{1}{\left[\frac{1}{{\mathrm{a}}_1}+\frac{1}{{\mathrm{a}}_2}\right]} \\ =\frac{{\mathbf{a}}_{\mathbf{1}}{\mathbf{a}}_{\mathbf{2}}}{{\mathbf{a}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}} \end{gathered}$$