Two like parallel forces P and Q act at the ends of a strianght ruler of length l. What is the distance by which the resultant of the forces will shift along the ruler If P is doubled ?

Answer

To solve the problem of how much the resultant of the forces will shift along the ruler when force P is doubled, we can follow these steps:

### Step 1: Understand the Setup
We have two parallel forces, P and Q, acting at the ends of a straight ruler of length \( l \). Let's assume force P acts at one end (let's say point A) and force Q acts at the other end (point B).

### Step 2: Calculate the Initial Position of the Resultant Force
The resultant force \( R \) of the two forces can be expressed as:
\[ R = P + Q \]

To find the position of the resultant force, we can use the concept of torque. We will take moments about point A (the point where force P acts). The torque due to force Q about point A is:
\[ \text{Torque due to Q} = Q \cdot l \]

The torque due to the resultant force \( R \) acting at a distance \( x \) from point A is:
\[ \text{Torque due to R} = R \cdot x = (P + Q) \cdot x \]

Setting the torques equal gives us:
\[ Q \cdot l = (P + Q) \cdot x \]

From this, we can solve for \( x \):
\[ x = \frac{Q \cdot l}{P + Q} \]

### Step 3: Consider the New Situation with P Doubled
Now, if we double the force P, the new force becomes \( 2P \). The new resultant force \( R' \) is:
\[ R' = 2P + Q \]

Using the same torque method, we set up the equation:
\[ Q \cdot l = (2P + Q) \cdot y \]

Where \( y \) is the new position of the resultant force. Solving for \( y \):
\[ y = \frac{Q \cdot l}{2P + Q} \]

### Step 4: Calculate the Shift in Position
The distance by which the resultant shifts is given by the difference in positions \( y \) and \( x \):
\[ \text{Shift} = |y - x| \]

Substituting the values of \( y \) and \( x \):
\[ \text{Shift} = \left| \frac{Q \cdot l}{2P + Q} - \frac{Q \cdot l}{P + Q} \right| \]

### Step 5: Simplify the Expression
To simplify the expression, we can find a common denominator:
\[ \text{Shift} = \left| \frac{Q \cdot l (P + Q) - Q \cdot l (2P + Q)}{(2P + Q)(P + Q)} \right| \]

This simplifies to:
\[ \text{Shift} = \left| \frac{Q \cdot l (P + Q - 2P - Q)}{(2P + Q)(P + Q)} \right| \]
\[ \text{Shift} = \left| \frac{-Q \cdot l (P)}{(2P + Q)(P + Q)} \right| \]

### Final Result
Thus, the distance by which the resultant of the forces will shift along the ruler when P is doubled is:
\[ \text{Shift} = \frac{Q \cdot l \cdot P}{(2P + Q)(P + Q)} \]