A uniform stick Ab 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

A uniform stick Ab 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

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Well, let's start balancing our way through this problem!

Since the stick is balanced when the weight is hung 10cm from one end, and then it is moved to 8.75cm, we know that the center of mass of the stick is somewhere between those two points.

To find the length of the stick, we can use the principle of moments. The moment of a force is found by multiplying the force by the perpendicular distance from the point of rotation (in this case, the knife edge).

So, when the weight is at 10cm from the knife edge, the total moment is 200N * 10cm = 2000Nm.

When the weight is at 8.75cm from the knife edge, the total moment is 200N * 8.75cm = 1750Nm.

Since the stick is balanced in both cases, the total moments must be the same. Therefore, we can set up the equation:

2000Nm = 1750Nm + W * x

Where W is the weight of the stick and x is the distance from the knife edge to the center of mass.

Now, since the stick is uniform, we can assume that its weight acts at its center. This means that the distance from the knife edge to the center of mass is half the length of the stick.

Let's call the length of the stick L. Therefore, x = L/2.

Now we can substitute this back into our equation:

2000Nm = 1750Nm + W * (L/2)

Solving for W:

250Nm = (W * L)/2

W * L = 500Nm

Now, let's find the weight of the stick. The weight is equal to the mass of the stick multiplied by the acceleration due to gravity (9.8m/s^2).

W = m * g

500Nm = m * 9.8m/s^2

m = 500Nm / 9.8m/s^2

m ≈ 51.02 kg

So, the weight of the stick is approximately 51.02 kg.

To find the length of the stick, we can rearrange our equation:

W * L = 500Nm

L = 500Nm / W

L ≈ 500Nm / 51.02 kg

L ≈ 9.8m

So, the length of the stick is approximately 9.8 meters.

To find the length of the stick and its weight, we can use the principle of moments. The principle of moments states that for an object to be in equilibrium, the sum of the anticlockwise moments about any point must be equal to the sum of the clockwise moments about the same point.

Let's take the knife edge as the reference point. We can analyze the two scenarios separately.

1. When the stick is balanced with the weight at one end:
In this scenario, the weight of the stick can be assumed to act at its center. Let's say the length of the stick is 'x'. We have the following equation:

(length of AB - distance of weight from knife edge) * weight of 200N = (distance of weight from knife edge) * weight of the stick

(10 cm - 0 cm) * 200 N = (0 cm) * weight of the stick

As the length of AB is 10 cm and the distance of the weight from the knife edge is 0 cm, the equation simplifies to:

200 N = 0 N

This equation is not possible, which means that the stick cannot be balanced without any weight.

2. When the stick is balanced with the weight at 8.75 cm from the knife edge:
In this scenario, we need to consider the weight of the stick and the weight of the 200N weight at opposite ends of the stick. Let the weight of the stick be 'W'. We have the following equation:

(length of AB - distance of weight from knife edge) * weight of 200N = (distance of weight from knife edge) * weight of the stick

(10 cm - 8.75 cm) * 200 N = (8.75 cm - 0 cm) * W

1.25 cm * 200 N = 8.75 cm * W

Multiplying both sides by 10 to eliminate decimal points:

12.5 cm * 200 N = 87.5 cm * W

2500 N cm = 87.5 cm * W

Dividing both sides by 87.5 cm:

(2500 N cm) / (87.5 cm) = W

W = 28.57 N

So, the weight of the stick is approximately 28.57 N.

To find the length of the stick, we can substitute this value of W into the previous equation:

2500 N cm = 87.5 cm * 28.57 N

Dividing both sides by 87.5 cm:

(2500 N cm) / (87.5 cm) = 28.57 N

x = 28.57 cm

Therefore, the length of the stick is approximately 28.57 cm.