A uniform meter rule PQ is balanced on a knife edge at the 80cm mark.If a mass of 200g is suspended at Q,calculate the mass of the meter rule.
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To solve this problem, we need to use the principle of moments. The principle states that for an object to be in equilibrium, the sum of the clockwise moments must equal the sum of the anticlockwise moments.
Let's assume that the mass of the meter rule is M grams. Given that the rule is balanced at the 80 cm mark, the distance of the center of gravity from the knife edge is also 80 cm.
The clockwise moment due to the 200 g mass at Q is given by Mq, where q is the distance of Q from the knife edge.
The anticlockwise moment due to the mass of the meter rule is given by Mr, where r is the distance of the center of gravity from the knife edge.
Since the rule is in equilibrium, the total clockwise moment equals the total anticlockwise moment:
Mq = Mr
We know that the distance Q is 100 cm from the knife edge (1 meter minus the 80 cm mark):
q = 100 cm
To find r, we need to consider the fact that the center of gravity of the meter rule is at its midpoint (M). So, r is half the length of the meter rule:
r = 80 cm / 2 = 40 cm
Now, we can set up the equation:
M(100 cm) = M(40 cm)
Simplifying the equation, we can divide both sides by M:
100 cm = 40 cm
Canceling out the common factor of M, we find:
100 = 40
Since this equation is false, it means that there is no mass M that can balance the meter rule. Therefore, the mass of the meter rule cannot be calculated with the given information.
To solve this problem, we can use the principle of moments, which states that for an object to be in equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments. In this case, the meter rule is balanced on the knife edge, so the total clockwise moments must be equal to the total anticlockwise moments.
Let's break down the problem step by step:
1. Determine the distance of the center of mass of the meter rule from the knife edge.
The meter rule is balanced at the 80 cm mark, which means the distance of the center of mass from the knife edge is also 80 cm.
2. Calculate the clockwise moment.
The clockwise moment is given by the product of the mass and the distance from the knife edge. In this case, the clockwise moment is (mass of meter rule) × (distance from the knife edge). Let's assume the mass of the meter rule as m.
Clockwise moment = m × 80 cm
3. Calculate the anticlockwise moment.
The only other object in the system is the 200 g mass suspended at Q. Since it is acting on the opposite side of the knife edge, its moment will be anticlockwise.
Anticlockwise moment = (mass of suspended object) × (distance from the knife edge to the suspended object)
Anticlockwise moment = 200 g × (80 cm - 100 cm) (Note: 80 cm - 100 cm gives the distance from the knife edge to the suspended object)
4. Set the clockwise moment equal to the anticlockwise moment.
Since the system is in equilibrium, the clockwise moment must be equal to the anticlockwise moment.
m × 80 cm = 200 g × (80 cm - 100 cm)
5. Convert units if necessary.
To have consistent units in the equation, convert the mass of the suspended object from grams to kilograms. 1 kg = 1000 g.
200 g = 0.2 kg
6. Solve the equation.
Substitute the known values into the equation and solve for m.
m × 80 cm = 0.2 kg × (80 cm - 100 cm)
m × 80 cm = -0.2 kg × (-20 cm)
m × 80 cm = 4 kg∙cm
m = 4 kg∙cm / 80 cm
m = 0.05 kg
Therefore, the mass of the meter rule is 0.05 kg, or 50 grams.