Subtopic: Design and selection of machine elements – gears, pulleys, chains and sprockets and belts
Question Statement:
An epicyclic gear train consists of a sun gear $S$ having $24$ teeth and a fixed internal ring gear $A$ having $72$ teeth. A planet gear $P$ meshes with both $S$ and $A$ and is carried on an arm $R$. If the sun gear rotates at $600\text{ rpm}$ clockwise, the rotational speed of the arm $R$ is ________ $\text{rpm}$ clockwise (answer in integer).
Answer Key:
150
Step-by-Step Solution & Explanation
Let the number of teeth on sun gear be $T_S = 24$ and internal ring gear be $T_A = 72$.
Using kinematic velocity relations for epicyclic train:
$\frac{N_A - N_R}{N_S - N_R} = -\frac{T_S}{T_A} = -\frac{24}{72} = -\frac{1}{3}$
Given $N_A = 0$ (fixed ring gear) and $N_S = +600\text{ rpm}$:
$\frac{0 - N_R}{600 - N_R} = -\frac{1}{3} \implies -3 N_R = -(600 - N_R) \implies 3 N_R = 600 - N_R \implies 4 N_R = 600 \implies N_R = 150\text{ rpm}$
Thus, the speed of the arm $R$ is $150\text{ rpm}$ clockwise.