A uniform meter stick of mass M has an empty paint can of mass m hanging from one end. The meter stick and the can balance at a point 19.9 cm from the end of the stick where the can is attached. When the balanced stick-can system is suspended from a scale, the reading on the scale is 2.49 N. Calculate the mass of the meter stick.Calculate the mass of the paint can.
To solve this problem, we can use the principle of balancing torques. The torque is the product of the force and the distance from the pivot point.
Let's define some variables:
M: mass of the meter stick (unknown)
m: mass of the paint can (unknown)
L: length of the meter stick
d: distance from the pivot point (attached end of the stick) to the balance point
Based on the information given, we know that the balance point is 19.9 cm (or 0.199 m) from the end of the stick where the can is attached.
Now, let's calculate the torque due to the weight of the meter stick:
Torque_meter stick = M * g * (L - d)
where g is the acceleration due to gravity (9.8 m/s^2).
The torque due to the weight of the paint can is given by:
Torque_paint can = m * g * d
Since the system is balanced, the two torques are equal:
Torque_meter stick = Torque_paint can
M * g * (L - d) = m * g * d
Now, let's calculate the mass of the meter stick (M):
M = (m * g * d) / (g * (L - d))
Given that d = 0.199 m and L = 1 m, we can substitute these values into the equation:
M = (m * 0.199) / (1 - 0.199)
Next, let's calculate the weight of the system using the scale reading:
Weight_system = 2.49 N
Since the scale reading in Newtons is the sum of all the weights, we can use this equation:
Weight_system = m * g + M * g
By substituting the value of M obtained above, we get:
2.49 = m * g + [(m * 0.199) / (1 - 0.199)] * g
Now, let's solve this equation to find the value of m (mass of the paint can):
Multiply both sides by (1 - 0.199):
2.49 * (1 - 0.199) = m * g * (1 - 0.199) + (m * 0.199) * g
2.49 * 0.801 = m * g * 0.801 + m * 0.199 * g
Divide both sides by g:
2.49 * 0.801 / g = m * 0.801 + m * 0.199
Now, let's simplify further:
1.998549 / g = m * 0.801 + m * 0.199
Subtract m * 0.199 from both sides:
1.998549 / g - m * 0.199 = m * 0.801
Rearrange the equation:
m * 0.801 + 0.199 * m = 1.998549 / g
Now, we have:
m * (0.801 + 0.199) = 1.998549 / g
m = 1.998549 / (g * 1)
Finally, substitute the value of g (9.8 m/s^2) to calculate the mass of the paint can (m):
m = 1.998549 / (9.8 * 1)
m ≈ 0.204 kg
Similarly, substitute the values of m, d, and L to calculate the mass of the meter stick (M):
M = (0.204 * 0.199) / (1 - 0.199)
M ≈ 0.0409 kg
Therefore, the mass of the meter stick is approximately 0.0409 kg, and the mass of the paint can is approximately 0.204 kg.