A metre ruler is balanced at the 35cm mark by a 1N weight placed at the 15cm mark. What is the mass of the ruler?
1 kg (35-15) = m(50-35)
m = 20/15 kg
Well, if the ruler is balanced, it must have a great sense of equilibrium! It's like the ruler has found its happy place in the middle. As for the mass, we know that the weight of the ruler is 1N. So, to find the mass, we need to divide that weight by the acceleration due to gravity, which is approximately 9.8 m/s². That would give us a mass of around 0.10 kilograms. So, the ruler is quite light, but don't be fooled by its weight – it can still measure things with precision!
To find the mass of the ruler, we need to consider the concept of torque or moment of a force. Torque is defined as the product of the force and the distance from the pivot point.
In this case, the 1N weight placed at the 15cm mark exerts a clockwise torque (since it is on the left side of the pivot) while the ruler itself exerts an equal and opposite anticlockwise torque (since it is on the right side of the pivot).
We can calculate the torque of the weight by multiplying the force (1N) by the distance from the pivot (15cm):
Torque of the weight = 1N * 15cm = 15 Ncm
Since the ruler is balanced and not rotating, the torque exerted by the weight is equal to the torque exerted by the ruler. Therefore, we can calculate the mass of the ruler by determining the distance from the pivot to its center of mass.
Let's assume the center of mass is located at the x cm mark from the pivot. The torque exerted by the ruler can be calculated by multiplying the weight (m * g, where m is the mass of the ruler and g is the acceleration due to gravity) by the distance from the pivot (35cm - x cm):
Torque of the ruler = (m * g) * (35cm - x cm)
Since the ruler is balanced, the torque exerted by the weight (15 Ncm) is equal to the torque exerted by the ruler:
15 Ncm = (m * g) * (35cm - x cm)
To simplify the calculations, let's assume the acceleration due to gravity g ≈ 10 m/s^2.
Simplifying the equation:
15 Ncm = (m * 10 m/s^2) * (35cm - x cm)
Divide both sides by 10m/s^2:
1.5 Ncm = m * (35cm - x cm)
To further simplify, we can convert cm to meters:
1.5 Ncm = m * (0.35m - 0.01x m)
Now we can substitute the values into the equation and solve for the mass of the ruler.
To find the mass of the ruler, we need to understand the concept of torque. Torque is the rotational equivalent of force, which is defined as the product of force and the perpendicular distance from the point of rotation.
In this scenario, the ruler is balanced, meaning that the sum of the torques on both sides of the balancing point is zero. Since the ruler is balanced at the 35cm mark, the torques on either side must be equal.
We know that the weight of the ruler is acting at the 15cm mark with a force of 1N. Therefore, the torque exerted by the weight is given by:
Torque = Force × Distance = 1N × 15cm
Now, for the ruler to be balanced, the torque exerted by the weight on one side must be equal to the torque exerted by the ruler itself on the other side. Since the ruler is balanced at the 35cm mark, the torque exerted by the ruler on the other side is given by:
Torque = Force × Distance = Mass × Gravity × Distance
Since the gravitational force is the same throughout the ruler, we can equate the torques:
1N × 15cm = Mass × Gravity × (35cm - 15cm)
Simplifying the equation:
15Ncm = Mass × Gravity × 20cm
Dividing both sides by 20cm:
0.75N = Mass × Gravity
Finally, solving for mass:
Mass = 0.75N / Gravity
Note: The value of gravity can vary depending on where you are on Earth. The standard value for gravity is approximately 9.8 m/s^2.