Principle of Moments (Cambridge International A-Level Physics)

The Principle of Moments

Spec point — 4.2.1

• The principle of moments states: For a system to be in equilibrium, the sum of clockwise moments about a point must be equal to the sum of the anticlockwise moments (about the same point)

Moments acting on a balanced beam

The sum of the anticlockwise moments of forces F₁and F₃are equal to the clockwise moment of force F₂

• In the above diagram:

  • Force F₂ is supplying a clockwise moment;
  • Forces F₁ and F₃ are supplying anticlockwise moments
  • The clockwise and anticlockwise moments are equal because the beam is in equilibrium

F_ 2d_ 2=(F_ 1d_ 1)+(F_ 3d_ 3)

Worked Example A uniform beam of weight 40 N is 5 m long and is supported by a pivot situated 2 m from one end.

When a load of weight W is hung from that end, the beam is in equilibrium as shown in the diagram.

What is the value of W?

A. 10 N

B. 50 N

C. 25 N

D. 30 N

Answer:

Step 1: List the known quantities

• Weight acting on beam, W_B = 40 N
• Weight acting on mass = W
• Length of beam = 5 m
• Distance to pivot (from end of beam), d_W = 2 m

Step 2: Recall the principle of moments

clockwise moments = anticlockwise moments

Step 3: Calculate the clockwise moment

• Because the beam is uniform, the force of weight acting upon it will be exerted from its centre of gravity
• This will be the middle of the beam

$\mathrm{Centre} \mathrm{of} \mathrm{gravity}=\frac{5}{2}=2.5m$ (from the end of the beam)

• The pivot is 2 m from the end of the beam
• Therefore, the force acts at a distance of 2.5 − 2 = 0.5 m from the pivot

$\mathrm{clockwise} \mathrm{moment}=W_{B}d$

$\mathrm{clockwise} \mathrm{moment}=40\times0.5$

$\mathrm{clockwise} \mathrm{moment}=20Nm$

Step 4: Calculate the anticlockwise moment

$\mathrm{anticlockwise} \mathrm{moment}=Wd_{W}$

$\mathrm{anticlockwise} \mathrm{moment}=W\times2=2W$

Step 5: Equate the clockwise and anticlockwise moments to calculate

$\mathrm{clockwise} \mathrm{moment}=\mathrm{anticlockwise} \mathrm{moment}$

$20=2W$

$W=\frac{20}{2}=10N$

• Therefore, the answer is A

Exam Hint Make sure that all the distances are in the same units and you’re considering the correct forces as clockwise or anticlockwise, as seen in the diagram below