A uniform meter rule of weight 1.0N, is pivoted at the 0.40m mark, A2.0N weight is hung from the 0.15m where must a 2.0N weight be placed to balance the meter rule
To balance the meter rule, the torque on each side of the pivot point must be equal. Torque is calculated by multiplying the force applied by the distance from the pivot point.
On one side of the pivot point, we have a 1.0N weight at a distance of 0.40m. So, the torque on this side is 1.0N * 0.40m = 0.40Nm.
On the other side, we have a 2.0N weight at a distance of 0.15m. Let's denote the distance from the pivot point where the 2.0N weight must be placed as "x". The torque on this side would be 2.0N * (0.15m - x).
Since the meter rule is balanced, the torque on both sides must be equal. Therefore, we can set up the equation:
0.40Nm = 2.0N * (0.15m - x)
Now solve for "x":
0.40Nm = 0.30N - 2.0Nx
0.40Nm + 2.0Nx = 0.30N
2.0Nx = 0.30N - 0.40Nm
2.0Nx = 0.30N - 0.40N(0.40m)
2.0Nx = 0.30N - 0.16N
2.0Nx = 0.14N
x = 0.14N / (2.0N)
x ≈ 0.07m
Therefore, the 2.0N weight must be placed at around the 0.07m mark to balance the meter rule.
To balance the meter rule, we can use the principle of moments. The principle of moments states that the sum of the anticlockwise moments is equal to the sum of the clockwise moments.
Let's assign the following variables:
- Mass of the uniform meter rule (M) = 1.0 kg (since weight = mass * acceleration due to gravity = 1.0N / 9.8 m/s² = 1.0 kg)
- Distance of the pivot point (0.40 m) from the 0 mark
- Mass hanging at the 0.15m mark (m₁) = 2.0 kg
- Mass that needs to be placed (m₂) to balance the meter rule
Now, we can calculate the clockwise and anticlockwise moments:
Clockwise moments: weight × distance
- Clockwise moment due to the 2.0N weight at the 0.15m mark = (2.0 N) × (0.15 m) = 0.30 Nm
- Clockwise moment due to the 2.0N weight at the unknown position (m₂) = (2.0 N) × (unknown distance from pivot)
Anticlockwise moments: weight × distance
- Anticlockwise moment due to the 1.0N weight of the meter rule at the 0.40m mark = (1.0 N) × (0.40 m) = 0.40 Nm
Since the meter rule is in equilibrium, the clockwise moments and anticlockwise moments must be equal:
0.30 Nm + (2.0 N) × (unknown distance) = 0.40 Nm
Simplifying the equation:
0.30 Nm + 2.0 N × (unknown distance) = 0.40 Nm
Rearranging the equation to solve for the unknown distance:
2.0 N × (unknown distance) = 0.40 Nm - 0.30 Nm
2.0 N × (unknown distance) = 0.10 Nm
Dividing both sides by 2.0 N:
(unknown distance) = 0.10 Nm / 2.0 N
(unknown distance) = 0.05 m
Therefore, to balance the meter rule, the 2.0N weight needs to be placed at a distance of 0.05 meters from the pivot point (A).
To determine where the 2.0N weight should be placed to balance the meter rule, we can use the principle of moments.
The principle of moments states that for an object to be in equilibrium (balanced), the sum of the clockwise moments must be equal to the sum of the anticlockwise moments.
In this case, the meter rule is pivoted at the 0.40m mark, and a 1.0N weight is located at that point. Let's call this point A.
We also have a 2.0N weight hanging from the 0.15m mark. Let's call this point B.
To balance the meter rule, we need to find the position of another 2.0N weight, which we can call point C.
The moment at any point is equal to the weight multiplied by the perpendicular distance from the point to the pivot.
Let's calculate the total clockwise and anticlockwise moments.
Clockwise moments:
moment of the 1.0N weight at A = 1.0N x 0.40m = 0.40Nm
Anticlockwise moments:
moment of the 2.0N weight at B = 2.0N x (0.15m - 0.40m) = 2.0N x (-0.25m) = -0.50Nm
Since the sum of the clockwise moments must be equal to the sum of the anticlockwise moments for the meter rule to be balanced, we have:
0.40Nm = -0.50Nm + moment of the 2.0N weight at C
To balance the meter rule, the sum of the clockwise and anticlockwise moments must be zero. Therefore, the moment of the 2.0N weight at C should be:
moment of the 2.0N weight at C = -0.40Nm + 0.50Nm
moment of the 2.0N weight at C = 0.10Nm
Now, to determine where the 2.0N weight should be placed, we can use the formula for moment: moment = weight x distance.
Given that the weight is 2.0N, and the moment we want is 0.10Nm, we can rearrange the formula to solve for the distance:
distance = moment / weight
distance = 0.10Nm / 2.0N
distance = 0.05m
Therefore, the 2.0N weight should be placed at the 0.05m mark from the pivot to balance the meter rule.