Question

THE LINE OF ACTION OF THE RESULTANT FORCE OF TWO LIKE PARALLEL FORCES SHIFTS BY ONE-FOURTH OF THE DISTANCE BETWEEN THE FORCES WHEN THE FORCES ARE INTERCHANGED. THE RATIO OF THE TWO FORCES IS ?

Answer 1

Answer on Question # 42027, Physics, Mechanics Kinematics | Dynamics

THE LINE OF ACTION OF THE RESULTANT FORCE OF TWO LIKE PARALLEL FORCES SHIFTS BY ONE-FOURTH OF THE DISTANCE BETWEEN THE FORCES WHEN THE FORCES ARE INTERCHANGED. THE RATIO OF THE TWO FORCES IS?

Solution

Let $P$ and $Q$ are two like parallel forces. The distance between $P$ and the resultant force is $\frac{d}{2} + a$, the distance between $Q$ and the resultant force is $\frac{d}{2} - a$, where $d$ is the distance between $Q$ and $P$. When the forces are interchanged the resultant force shifts symmetrically to the middle of the distance between $Q$ and $P$ by $2a = \frac{1}{4} d$. So $a = \frac{1}{8} d$. We have that

$$\frac{P}{\frac{d}{2} + a} = \frac{Q}{\frac{d}{2} - a} \rightarrow \frac{P}{\frac{d}{2} + \frac{1}{8} d} = \frac{Q}{\frac{d}{2} - \frac{1}{8} d} \rightarrow \frac{P}{\frac{5}{8} d} = \frac{Q}{\frac{3}{8} d}.$$

The ratio of the forces is

$$\frac{P}{Q} = \frac{\frac{5}{8} d}{\frac{3}{8} d} = \frac{5}{3}.$$

Answer: $\frac{5}{3}$

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