A uniform half metre rule is freely pivoted at the 20cm mark and it balances horizontally when a weight of 10N is suspended at 4cm mark, calculate weight of the meter rule and the reaction of the knife
down
10 N at (20-4) = 16 cm left from pivot
m g N at (50-20) = 30 cm right of pivot
so 10 *16 = mg * 30
mg = 16/3 = 5.33 Newtons
force up = 10 + 5.33 Newtons
To solve for the weight of the meter rule and the reaction of the knife, we can use the principle of moments.
Let's define the following variables:
- W: weight of the meter rule
- R: reaction at the knife edge
- d1: distance from the pivot to the weight (4cm mark)
- d2: distance from the pivot to the center of gravity of the meter rule (50cm mark)
- d3: distance from the pivot to the knife edge (20cm mark)
To balance a meter rule horizontally, the sum of the clockwise moments should be equal to the sum of the anticlockwise moments. In other words, the product of the weight and its distance from the pivot (moment) should be the same on both sides.
Let's calculate the moments:
Clockwise moments:
Moment due to the weight = 10N × d1
Anticlockwise moments:
Moment due to the weight of the meter rule = W × d2
Moment due to the reaction at the knife edge = R × d3
Since the meter rule is balanced, the clockwise moments are equal to the anticlockwise moments:
10N × d1 = W × d2 + R × d3
Given that d1 = 4cm, d2 = 50cm, and d3 = 20cm, we can substitute these values into the equation:
10N × 4cm = W × 50cm + R × 20cm
Simplifying the equation:
40Ncm = 50W + 20R
Since we have two unknowns, we need one more equation to solve the system of equations. We can use the fact that the meter rule is freely pivoted to balance horizontally. This implies that the sum of the vertical forces should be zero:
W + R = 10N
Now we have two equations:
40Ncm = 50W + 20R
W + R = 10N
We can solve this system using simultaneous equations. Subtracting the second equation from the first equation:
40Ncm - (W + R) = 50W + 20R - (W + R)
40Ncm - 10N = 50W + 20R - W - R
40Ncm - 10N = 49W + 19R
Simplifying further:
40N - 10N = 49W + 19R
30N = 49W + 19R
Now we have a simplified equation that contains only W and R as unknowns.
Unfortunately, we don't have enough information to find the specific values of W and R without additional data. To solve for W and R, we would need either the value of W or R, or another equation that relates them.
To solve this problem, we can use the principle of moments.
The principle of moments states that for an object to be in equilibrium (balanced and not rotating), 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, we will take moments about the pivot point, which is at the 20cm mark.
Let's denote the weight of the meter rule as W1 and the reaction of the knife as R.
Clockwise moments:
Moment due to the weight at the 4cm mark = 10 N * (20 cm - 4 cm) = 10 N * 16 cm (clockwise)
Anticlockwise moments:
Moment due to the weight of the meter rule at its center (50 cm mark) = W1 * (50 cm - 20 cm) = W1 * 30 cm (anticlockwise)
Moment due to the reaction of the knife at the pivot (20 cm mark) = R * (20 cm - 20 cm) = 0 (no moment)
Since the meter rule is balanced, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments:
10 N * 16 cm = W1 * 30 cm
Now we can solve this equation to find the weight of the meter rule (W1).
10 N * 16 cm = W1 * 30 cm
Divide both sides of the equation by 30 cm:
(10 N * 16 cm) / 30 cm = W1
The cm units cancel out:
(10 N * 16) / 30 = W1
Now we can solve this equation using a calculator:
(10 N * 16) / 30 ≈ 5.33 N
So, the weight of the meter rule is approximately 5.33 N.
To find the reaction of the knife (R), we can use the fact that the sum of the vertical forces must be zero since the meter rule is in equilibrium. This means that the upward reaction force (R) must balance the downward weight of the meter rule (W1).
Therefore, R = W1 = approximately 5.33 N.
So, the reaction of the knife is approximately 5.33 N.