A unifom pencil AB weighing 40g can be balanced horizontally on a knife adge at 2cm from the end A when a mass of 60g is hung from this end. What is the length of pencil?
Let the length of the pencil be x cm.
Since the pencil can be balanced horizontally on a knife edge at 2 cm from end A,
The moment about the knife edge due to the weight of the pencil (40g) is balanced by the moment about the knife edge due to the weight hanging from end A (60g).
The moment due to the weight of the pencil is given by: 40g * (x - 2) cm.
The moment due to the weight hanging from end A is given by: 60g * 2 cm.
Setting the two moments equal, we have:
40g * (x - 2) = 60g * 2.
Simplifying the equation:
40(x - 2) = 120,
40x - 80 = 120,
40x = 200,
x = 5.
Therefore, the length of the pencil is 5 cm.
To solve this question, 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.
Let's consider the moments about the knife edge at point A. The pencil has a weight of 40g acting downwards at its center of gravity. The distance "d1" from the knife edge at A to the center of gravity can be calculated by subtracting 2cm from the total length of the pencil.
Using the principle of moments, the clockwise moment is given by:
Clockwise moment = Weight of pencil (40g) × Distance from A (d1)
Now, a mass of 60g is hung from end A. The distance between the knife edge at A and the mass is given as 2cm. The anticlockwise moment is given by:
Anticlockwise moment = Mass (60g) × Distance from A (2cm)
Since the pencil is balanced, the sum of the clockwise moments is equal to the sum of the anticlockwise moments:
Clockwise moment = Anticlockwise moment
40g × d1 = 60g × 2cm
Now, let's solve for the length of the pencil (d1):
40g × d1 = 60g × 2cm
To find the length in cm, divide both sides of the equation by 40g:
d1 = (60g × 2cm) / 40g
Simplifying:
d1 = 120g cm / 40g
d1 = 3cm
Since d1 is the distance from the knife edge at A to the center of gravity, to find the total length of the pencil, we need to add the 2cm distance between the knife edge and the mass, as well as the distance from the center of gravity to end B.
Total length of the pencil = d1 + 2cm + distance from center of gravity to end B
However, since the pencil is described as a uniform pencil, the center of gravity is located at the midpoint. Therefore, the distance from the center of gravity to end B is equal to the distance from the center of gravity to A, which is d1.
Thus, the total length of the pencil is:
Total length = d1 + 2cm + d1
Total length = 3cm + 2cm + 3cm
Total length = 8cm
Therefore, the length of the pencil is 8cm.
To solve this problem, we can use the principle of moments:
The principle of moments 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 clockwise moment is caused by the weight of the pencil (40g) acting at the center of the pencil, while the anticlockwise moment is caused by the weight hanging from the end of the pencil (60g) acting at a distance of 2cm from end A.
Let's assume that the length of the pencil is 'x' cm.
The clockwise moment can be calculated as 40g * (x/2) cm, as the weight of the pencil is acting at the midpoint.
The anticlockwise moment can be calculated as 60g * (x - 2) cm, as the weight hanging from the end is acting at a distance of x - 2 cm from end A.
According to the principle of moments, these two moments should be equal for the pencil to be in equilibrium.
40g * (x/2) = 60g * (x - 2)
Now, let's solve this equation:
40g * (x/2) = 60g * (x - 2)
20g * x = 60g * (x - 2)
20g * x = 60g * x - 120g
120g - 20g * x = 60g * x
120g = 80g * x
x = 120g / 80g
x = 1.5 cm
Therefore, the length of the pencil is 1.5 cm.