A uniform metre rule of mass 90g is pivoted at the 40cm mark. If the rule is in equilibrium with an unknown mass market placed at the 10cm mark and a 72g at the 70cm mark,determine m.
To determine the unknown mass (m), we need to consider the principle of moments. The principle of moments states that for an object to be in equilibrium, the sum of the anticlockwise moments must be equal to the sum of the clockwise moments.
In this scenario, the meter rule is in equilibrium, so the sum of the anticlockwise moments is equal to the sum of the clockwise moments.
First, let's determine the moment caused by the unknown mass (m):
The moment (M) caused by a force (F) acting at a distance (d) from the pivot point is given by the equation: M = F × d
In this case, the force is the weight of the unknown mass (m), which is given by the equation: F = m × g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The distance (d) for the unknown mass is 40 cm (since it is placed at the 10 cm mark).
So, the moment caused by the unknown mass (M1) is:
M1 = (m × g) × (0.4 m)
Now let's determine the moment caused by the 72 g mass:
The force (F2) exerted by the 72 g mass is given by: F2 = (72 g) × g
The distance (d2) for the 72 g mass is 30 cm (since it is placed at the 70 cm mark).
So, the moment caused by the 72 g mass (M2) is:
M2 = (72 g × g) × (0.3 m)
Since the object is in equilibrium, M1 = M2.
Now we'll substitute the values and solve for m:
(m × g) × (0.4 m) = (72 g × g) × (0.3 m)
Now divide both sides of the equation by (g × 0.4) to solve for m:
m = (72 g × g × 0.3 m) / (g × 0.4)
Canceling out the terms, we get:
m = (72 g × 0.3 m) / 0.4
Multiply the numerators:
m = 21.6 g m / 0.4
Simplify:
m = 54 g
Therefore, the unknown mass (m) is 54 g.