Moments, Equilibrium and Centre of Gravity: Question 4

Worked solution

Part (a): Conditions for equilibrium

For the plank to be in equilibrium, both of the following must be true:

• The resultant force acting on the plank is zero.
• The resultant moment acting on the plank (about any point) is zero.

Both conditions are needed: a zero resultant force alone would not stop the plank rotating, and a zero resultant moment alone would not stop it accelerating sideways.

Part (b): Taking moments about A

Taking moments about A has an advantage: the support force at A, $R_A$, acts at A itself, so its perpendicular distance from A is zero and it produces no moment about A. This means $R_A$ drops out of the equation entirely.

The plank is uniform, so its weight acts at its centre, which is $\dfrac{4.0}{2} = 2.0\text{ m}$ from A:

$\text{moment of plank's weight about A} = 200\text{ N} \times 2.0\text{ m} = \boxed{400\text{ N m}}$

The person’s weight acts $1.0\text{ m}$ from A:

$\text{moment of person's weight about A} = 600\text{ N} \times 1.0\text{ m} = \boxed{600\text{ N m}}$

Both of these moments turn the plank the same way about A (they both tend to make it rotate downward on the side away from A), so they add together. This must be balanced by the opposing moment of the support force at B, which acts $4.0\text{ m}$ from A:

$R_B \times 4.0\text{ m} = 400\text{ N m} + 600\text{ N m} = 1000\text{ N m}$

$R_B = \frac{1000\text{ N m}}{4.0\text{ m}} = \boxed{250\text{ N}}$

Part (c): Using the resultant force condition

Since the resultant force on the plank is zero, the two support forces acting upward must together balance the total weight acting downward (the plank’s weight plus the person’s weight):

$R_A + R_B = 200\text{ N} + 600\text{ N} = 800\text{ N}$

$R_A = 800\text{ N} - R_B = 800\text{ N} - 250\text{ N}$

$R_A = \boxed{550\text{ N}}$

Part (d): Moving further from A

As the person walks further from A (closer to B), the perpendicular distance between her weight and A increases. This increases her moment about A. Since the distance from A to B is fixed at $4.0\text{ m}$, a bigger moment about A must now be balanced by a bigger moment from $R_B$, which means $R_B$ itself must increase.

Because $R_A + R_B$ must always equal the constant total weight of $800\text{ N}$, if $R_B$ increases then $R_A$ must correspondingly decrease. This matches what is physically reasonable: as the person moves closer to support B, more of her weight is supported by B and less by A.

Final answers

• (a) Resultant force $=$ zero and resultant moment $=$ zero
• (b) Moment of plank’s weight about A $=$ $400\text{ N m}$; moment of person’s weight about A $=$ $600\text{ N m}$; $R_B =$ $250\text{ N}$
• (c) $R_A =$ $550\text{ N}$
• (d) $R_B$ increases (and $R_A$ decreases), because the person’s moment about A increases as her distance from A increases