A meter stick is found to balance at the 47.3cm mark when placed on a fulcrum. When a 45.7g mass is attached at the 10.5cm mark, the fulcrum must be moved to the 38.5cm mark for balance. What is the mass of the meter stick?
The mass of meter stick= 151.77g but with 3 sig figs it's roughly 152grams
(47.7g)*(38.5cm-10.5cm)=(47.3cm-38.5cm)*(Mass of meter stick)
Simple algebra given the information, hope this helps!
To solve this problem, we need to understand the concept of torque. Torque is the product of force and the perpendicular distance from the fulcrum to the line of action of the force.
Let's denote the mass of the meter stick as M and its length as L. The torque exerted by the meter stick on the fulcrum can be calculated as follows:
Torque 1 = M * g * d1
where g is the acceleration due to gravity and d1 is the distance of the fulcrum from the center of mass of the meter stick.
Similarly, when the 45.7g mass is attached at the 10.5cm mark, the torque exerted by this mass on the fulcrum can be calculated as:
Torque 2 = 0.0457 kg * g * (L/2 - 10.5 cm)
Since the system is in balance, the torques exerted by the meter stick and the attached mass must be equal. Therefore, we can set up the following equation:
M * g * d1 = 0.0457 kg * g * (L/2 - 10.5 cm)
Now, we are given that the fulcrum must be moved to the 38.5cm mark for balance. This means the distance from the fulcrum to the center of mass of the meter stick changes. Let's denote this new distance as d2.
Applying the principle of torque balance again, we can set up the following equation:
M * g * d2 = 0.0457 kg * g * (L/2 - 38.5 cm)
Since the mass and gravity are common factors in both equations, we can cancel them out:
d1 = 0.0457 kg * (L/2 - 10.5 cm) (equation 1)
d2 = 0.0457 kg * (L/2 - 38.5 cm) (equation 2)
Now we can solve these two equations simultaneously to find the values of d1 and d2:
d1 = 0.0457 kg * (L/2 - 10.5 cm)
d2 = 0.0457 kg * (L/2 - 38.5 cm)
To make the calculation easier, let's divide equation 1 by equation 2:
d1 / d2 = (L/2 - 10.5 cm) / (L/2 - 38.5 cm)
Cross multiplying this equation leads to:
d1 * (L/2 - 38.5 cm) = d2 * (L/2 - 10.5 cm)
Now, substitute the values of d1 and d2 from their respective equations:
0.0457 kg * (L/2 - 10.5 cm) * (L/2 - 38.5 cm) = 0.0457 kg * (L/2 - 38.5 cm) * (L/2 - 10.5 cm)
Simplifying this equation will help determine the solutions for L, which is the length of the meter stick.
Once you find the value of L, you can substitute it back into either equation 1 or equation 2 to determine the distance (d1 or d2) from the fulcrum to the center of mass of the meter stick. Finally, you can calculate the mass (M) of the meter stick using either d1 or d2 in either of the original equations.