A uniform pencil AB weighing 40g can be balanced horizontally
on a knife edge at 2cm from
the end A when a mass of 60g is hung from this end. What is
the length of a pencil?
Well, you know what they say about balancing acts - it's all about finding the right center of gravity! So, let's do some circus calculations here.
We have a uniform pencil AB, and it's balanced horizontally on a knife edge at 2cm from end A. When a 60g mass is hung from end A, it manages to stay balanced. Now, we want to find the length of the pencil.
Since the pencil is balanced, we can assume that the center of gravity is right at the knife edge. This means that the total weight of the pencil on one side must be equal to the weight on the other side.
On one side, we have the pencil itself weighing 40g, and on the other side, we have the 60g mass hanging from it. So, the total weight on each side is the same.
Since the distance from the knife edge to end A is 2cm, and the weight of the pencil on each side must be equal, we can set up a proportion:
40g / 2cm = 60g / x
Solving for x, we find that x = (2cm * 60g) / 40g = 3cm
So, the length of the pencil AB is 3cm. Just remember, always keep your balance, both in life and in pencil acrobatics!
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 about any point must be equal to the sum of the anticlockwise moments about the same point.
In this case, the knife edge acts as the point of equilibrium. Let's denote the length of the pencil as 'l' and the distance between the knife edge and point A as 'x' (which is given as 2 cm or 0.02 m).
The clockwise moment about the knife edge is given by the weight of the pencil (40 g) multiplied by the distance between the knife edge and the center of gravity of the pencil (l/2). This can be written as 40g * (l/2).
The anticlockwise moment is given by the weight (60 g) multiplied by the distance between the knife edge and the hanging mass (2 cm or 0.02 m). This can be written as 60g * (0.02 m).
Since the pencil is balanced horizontally, the sum of the clockwise moments and anticlockwise moments must be equal. Therefore, we can set up an equation:
40g * (l/2) = 60g * (0.02 m)
Now, let's solve this equation for 'l', the length of the pencil:
40g * (l/2) = 60g * (0.02 m)
Divide both sides of the equation by 40g:
(l/2) = 60g * (0.02 m) / 40g
Simplify the expression:
(l/2) = 0.03 m
Multiply both sides of the equation by 2:
l = 2 * 0.03 m
l = 0.06 m
Therefore, the length of the pencil is 0.06 m.
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.
First, let's calculate the clockwise and anticlockwise moments.
The weight of the pencil can be considered as acting at its midpoint, which is 2 cm from point A.
Clockwise moment = weight of pencil x distance to knife edge
= 40g x 2cm
Anticlockwise moment = weight hung x distance to knife edge
= 60g x (length of pencil - 2cm)
Given that the pencil is balanced horizontally, the clockwise moment is equal to the anticlockwise moment.
40g x 2cm = 60g x (length of pencil - 2cm)
Now, let's solve this equation to find the length of the pencil.
80cm = 60g x (length of pencil - 2cm)
Divide both sides of the equation by 60g:
80cm / 60g = length of pencil - 2cm
Simplifying further:
4/3 cm/g = length of pencil - 2cm
Add 2cm to both sides:
length of pencil = 4/3 cm/g + 2cm
Therefore, the length of the pencil is 4/3 cm/g + 2cm.