A meter stick is found to balance at the 49.7-cm mark when placed on a fulcrum. When a 70.0-gram mass is attached at the 16.0-cm mark, the fulcrum must be moved to the 39.2-cm mark for balance. What is the mass of the meter stick?
Answer in grams
To find the mass of the meter stick, we can use the principle of moments. The principle of moments states that the sum of the clockwise moments is equal to the sum of the counterclockwise moments when an object is in equilibrium.
First, let's define the variables:
M = mass of the meter stick (what we need to find)
d1 = distance of the 70.0-gram mass from the fulcrum = 16.0 cm
d2 = distance of the meter stick's center of mass from the fulcrum = unknown
d3 = distance of the 49.7-cm mark from the fulcrum = 49.7 cm
d4 = distance of the 39.2-cm mark from the fulcrum = 39.2 cm
Now let's set up the equation based on the principle of moments:
Clockwise moment: 70.0 g × d1
Counterclockwise moment: M × d2 + M × d3
Since the meter stick is balanced, the moments are equal, so we can set up an equation:
Clockwise moment = Counterclockwise moment
70.0 g × d1 = M × d2 + M × d3
Substituting the known values:
70.0 g × 16.0 cm = M × d2 + M × 49.7 cm
Now we can solve for the mass of the meter stick:
70.0 g × 16.0 cm = M × (d2 + 49.7 cm)
1120 g·cm = M × (d2 + 49.7 cm)
Next, let's consider the second scenario where the fulcrum is moved. The principle of moments still applies:
Clockwise moment: 70.0 g × d1
Counterclockwise moment: M × d4 + M × d3
Again, the moments are equal, so we can set up another equation:
70.0 g × d1 = M × d4 + M × d3
Substituting the known values:
70.0 g × 16.0 cm = M × 39.2 cm + M × 49.7 cm
Simplifying:
1120 g·cm = M × (39.2 cm + 49.7 cm)
1120 g·cm = M × 88.9 cm
Now we have two equations with the same left-hand side:
1120 g·cm = M × (d2 + 49.7 cm)
1120 g·cm = M × 88.9 cm
Since both equations are equal to 1120 g·cm, we can equate the right-hand sides:
M × (d2 + 49.7 cm) = M × 88.9 cm
We can cancel out the mass (M) from both sides since it is common:
d2 + 49.7 cm = 88.9 cm
Now, we can solve for d2 by subtracting 49.7 cm from both sides:
d2 = 88.9 cm - 49.7 cm
d2 = 39.2 cm
Therefore, the distance of the meter stick's center of mass from the fulcrum is 39.2 cm.
Now we can substitute this value into either of the original equations to solve for the mass of the meter stick. Let's use the first equation:
70.0 g × 16.0 cm = M × d2 + M × 49.7 cm
Plugging in the known values:
70.0 g × 16.0 cm = M × 39.2 cm + M × 49.7 cm
Now we can solve for M:
1120 g·cm = M × (39.2 cm + 49.7 cm)
1120 g·cm = M × 88.9 cm
Dividing both sides by 88.9 cm:
M = 1120 g·cm / 88.9 cm
M ≈ 12.59 grams
Therefore, the mass of the meter stick is approximately 12.59 grams.