A plank AB has length 6m and mass 30kg - Edexcel - A-Level Maths Mechanics - Question 3 - 2017 - Paper 1

To solve for the value of $x$, we will apply the principle of moments around point A:

1. Identify the forces acting on the plank:

   • Weight of the plank: $30g$ acting downwards at its center (3m from A).
   • Weight of each person: $75g$ acting downwards at points P ($x$ m from A) and Q ($2x$ m from A).

2. Set up the moments about point A:

   $R + 5R = 75g \times x + 30g \times 3 + 75g \times 2x$

   Thus, we have: $6R = 75gx + 30g \times 3 + 75g \times 2x$

3. Since the plank is in equilibrium, we can also write:

   $M(4) = 75g \times 2x + 30g \times 3 - 5R = 0$

   This leads to: $75g \times 2x + 30g \times 3 = 5R$

4. Solve the equations simultaneously to find the value of $x$. From the moments equation, substituting back we get:

   $M(4) = 75gx + 75g \times 2x + 30g \times 3 = 5R \cdots (1)$

   By balancing moments:

   $x = \frac{2.3}{3} = 0.7667 m$

   Thus, the final value of $x$ is approximately 0.77 m.