A uniformed stick AB can be balanced on a knife edge 10cm from one end when a weight of 200N is hang from that end. When the knife edges is removed 5cm further from the end. The 200N weight has to be removed to a point 8.75cm from the knife edge to obtain a balance. Find the length of the stick and its weight.

Question

A uniformed stick AB can be balanced on a knife edge 10cm from one end when a weight of 200N is hang from that end. When the knife edges is removed 5cm further from the end. The 200N weight has to be removed to a point 8.75cm from the knife edge to obtain a balance. Find the length of the stick and its weight.

Answer 1

To solve this problem, we can use the principle of moments. The principle of moments states that for an object to be in rotational equilibrium, the sum of the clockwise moments about any point must be equal to the sum of the anticlockwise moments about that point.

Let's denote the length of the stick as L and its weight as W. We'll also assume that the center of gravity of the stick is at its midpoint.

First, let's calculate the clockwise and anticlockwise moments about the knife edge when the stick is balanced with a weight of 200N hanging 10cm from one end.

Clockwise moment = Weight x Distance from the knife edge
= 200N x 10cm = 2000 N-cm

Anticlockwise moment = Stick weight x Distance from the knife edge
= W x (L - 10cm)

Since the stick is balanced, the clockwise moment is equal to the anticlockwise moment:
2000 N-cm = W x (L - 10cm) ...(Equation 1)

Now, let's calculate the clockwise and anticlockwise moments when the knife edge is removed 5cm further from the end, and the stick is balanced with the weight of 200N placed 8.75cm from the knife edge.

Clockwise moment = Weight x Distance from the knife edge
= 200N x 15cm = 3000 N-cm

Anticlockwise moment = Stick weight x Distance from the knife edge
= W x (L - 15cm)

Again, since the stick is balanced, the clockwise moment is equal to the anticlockwise moment:
3000 N-cm = W x (L - 15cm) ...(Equation 2)

We now have a system of two equations (Equation 1 and Equation 2) with two unknowns (L and W). We can solve this system to find the values of L and W.

Divide Equation 2 by Equation 1:

(3000 N-cm) / (2000 N-cm) = (W x (L - 15cm)) / (W x (L - 10cm))

Simplifying, we get:

1.5 = (L - 15cm) / (L - 10cm)

Cross multiplying, we have:

1.5(L - 10cm) = L - 15cm

Simplifying further:

1.5L - 15cm = L - 15cm

Subtracting L from both sides:

0.5L = 0

Therefore, L = 0, which is not possible since the stick cannot have zero length. This means that the system of equations is inconsistent. There is no solution for the length of the stick and its weight that satisfies both conditions.

Hence, there is no valid solution to this problem.