A uniform stick can be nalanced on a knife edge 10cm from one end when a weight of 200N is hung from that end the 200N weight has to be moved to a point 8.75cm from the knife edge to obtain a balance .Find the length of the stick and its weight.
To solve this problem, we can use the principle of moments. The principle of moments states that for an object 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.
Let's denote the length of the stick as L and its weight as W. We'll also label the distance from the knife edge to the weight as x.
Given that a weight of 200N hangs 10cm from one end, we can set up an equation using the principle of moments:
(200N)(10cm) = (W)(L)(x)
Simplifying the equation, we get:
2000 N.cm = W.L.x
Now, we're also given that when the weight is moved to a point 8.75cm from the knife edge, the stick is balanced. This means that the clockwise and anticlockwise moments are equal. We can set up another equation:
(W)(L)(8.75cm) = (W)(L)(L - 8.75cm)
Simplifying this equation, we get:
8.75cm(L) = L(L - 8.75cm)
Expanding and rearranging the equation:
8.75L = L^2 - 8.75L
L^2 - 17.5L = 0
L(L - 17.5) = 0
So, we have two possible solutions for L: L = 0 or L = 17.5.
Since the length of the stick cannot be zero, we choose the solution L = 17.5 cm.
Now, we can substitute this value of L into the first equation to find the weight W:
2000 N.cm = W(17.5cm)(x)
W = 2000 N.cm / (17.5cm)(x)
Given that x is now 8.75cm, we can substitute this value and solve for W:
W = 2000 N.cm / (17.5cm)(8.75cm)
W ≈ 13.714 N
Therefore, the length of the stick is 17.5 cm and its weight is approximately 13.714 N.