Question

A uniform metre rule of mass 100g is balanced on a fulcrum at mark 40 cm by suspending an unknown mass m at the mark 20 cm . Find the value of m .To which side the rule will tilt if the mass m is moved to the mark 10 cm ? What is the resultant moment now ? How can it be balanced by another mass 50 g?

Answer 1

Mass of the ruler = 100g = 0.1 kg 

Length of the ruler l = 1m 

Moment of the unknown mass of the fulcrum is equal to mg(0.2) 

Moment of the mass of the ruler of the fulcrum is equal to (0.1)g(0.1) 

Because the system is in equilibrium so mg(0.2) = (0.1)g(0.1) 

=> m = 0.05kg = 50g 

 

The 50g mass is moved to mark 10cm 

The moment of the fulcrum because of the above stated mass is (0.05)(g)(0.3) 

= 0.147 Nm 

The movement of the ruler about the fulcrum is (0.1)g(0.1) = 0.098 Nm 

The ruler will tilt towards the mass placed at 10cm mark 

To make the system equal let another mass 50g or 0.05kg be placed 

at xm from the fulcrum 

 

Moment of the 50g mass at 10 cm mark is equal to moment of the ruler about the fulcrum + moment of the new 50g mass about the fulcrum 

=>0.147 = 0.098 + (0.05)(g)(x) 

=> 0.049 = 0.49x 

=> x = 0.1m = 10cm from the fulcrum 

Thus, the new 50g is placed at 50cm mark of the ruler

Answer 2

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From the principle of moments,

Clockwise moment = Anticlockwise moment

100 g × (50 – 40) cm = m × (40 – 20) cm

100 g × 10 cm = m × 20 cm

m = 50 g

If the mass m is moved to the mark 10 cm, the rule will tilt on the side of mass m (anticlockwise)

Anticlockwise moment if mass m is moved to the mark 10 cm

= 50 g × (40 – 10) cm

= 50 g × 30 cm

= 1500 g cm

Clockwise moment = 100 g × (50 – 40) cm

= 100 g × 10 cm

= 1000 g cm

Resultant moment = 1500 g cm – 1000 g cm

= 500 g cm (anticlockwise)

According to the principle of moments.

Clockwise moment = Anticlockwise moment

To balance it, 50 g weight should be kept on right-hand side so as to produce a clockwise moment. Let d cm be the distance from the fulcrum. Then,

100 g × (50 – 40) cm + 50 g × d = 50 g × (40 – 10) cm

100 g × 10 cm + 50 g × d = 50 g × 30 cm

1000 g cm + 50 g × d = 1500 g cm

50 g × d = 500 g cm

Then, d = 10 cm

It can be balanced by suspending the mass 50 g at the mark 50 cm.

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