Suppose you need to measure the mass of an object, but have only a 7-kg standard mass and a meter stick. Place a knife edge under the 50-cm mark of the meter stick. Then put the 7-kg standard mass at the 39.1-cm mark and place the object so as to balance the standard mass. If the system is balanced when the object is at the 85.9-cm mark, find the unknown mass.
7*10.9 = M*35.9
Solve for M
The two numbers above are the distances of the masses from the fulcrum at the 50 cm midpoint location.
To find the unknown mass, we can apply the principle of moments or rotational equilibrium. In rotational equilibrium, the sum of the clockwise moments is equal to the sum of the anticlockwise moments.
In this scenario, we have three masses: the 7-kg standard mass, the unknown mass, and the meter stick (which can be considered as a mass concentrated at its center).
Let's assign some variables:
- Mass of the unknown object: M (in kg)
- Distance from the knife edge to the 7-kg mass: x (in cm)
- Distance from the knife edge to the unknown mass: y (in cm)
- Length of the meter stick: L (in cm)
From the given information, we have:
- Distance from the knife edge to the standard 7-kg mass: 39.1 cm
- Distance from the knife edge to the balance point with the unknown mass: 85.9 cm
- Length of the meter stick: 100 cm (assumed)
To apply the principle of moments, let's set up an equation:
Clockwise moments = Anticlockwise moments
(7 kg) * (39.1 cm - x) = (M kg) * (85.9 cm - y)
Now let's solve for the unknown mass, M:
7 * (39.1 - x) = M * (85.9 - y)
To simplify the equation, we need to convert the length units from centimeters (cm) to meters (m) because mass is typically measured in kilograms (kg):
7 * (0.391 - x * 0.01) = M * (0.859 - y * 0.01)
Simplifying further:
2.737 - 7x = 0.859M - My
Now, we need another equation using the fact that the system is balanced:
Clockwise moments = Anticlockwise moments
(7 kg) * (39.1 cm - x) + (M kg) * (85.9 cm - y) = (Length of the meter stick, L) * (Weight of the meter stick)
To calculate the weight of the meter stick, we can assume it is balanced at its center. Therefore, the weight acts at the midpoint.
(Weight of the meter stick) = (Mass of the meter stick) * (Gravity) * (Distance from the knife edge to the midpoint)
Now, let's substitute the values:
(7 kg) * (39.1 cm - x) + (M kg) * (85.9 cm - y) = (L kg) * (9.8 m/s^2) * (L/2) * (0.01 m/cm)
Substituting the given values:
7 * (39.1 - x) + M * (85.9 - y) = 1 * (9.8) * (100/2) * (0.01)
Simplifying further:
(7 * 39.1 - 7 * x) + (M * 85.9 - M * y) = 4.9
273.7 - 7x + 85.9M - My = 4.9
Simplifying further:
-7x + 85.9M - My = 4.9 - 273.7
-7x + 85.9M - My = -268.8
We now have a system of two equations:
1) -7x + 85.9M - My = -268.8
2) 2.737 - 7x = 0.859M - My
We can solve these equations simultaneously to find the values of x and M. Once we have the value of M, it will be the unknown mass in kilograms.