Suppose the ruler in procedure 3 is asymmetrical, balancing at the 60.2 cm mark. The ruler is now supported at the 41.3 cm mark, and a mass of 364 g is placed at the 26.2 cm mark.
Find the mass of the ruler.
T=rF
To find the mass of the ruler, we can use the principle of torque. Torque is defined as the product of the force applied and the distance from the point of rotation. In this case, the point of rotation is the 60.2 cm mark.
The torque equation is given by:
T = r * F
Where:
T = Torque
r = Distance from the point of rotation
F = Applied force
In procedure 3, the ruler is supported at the 41.3 cm mark and a mass of 364 g is placed at the 26.2 cm mark.
To balance the ruler, the torques on both sides of the ruler must be equal.
On the left side of the ruler (from the point of rotation):
T_left = r_left * F_left
On the right side of the ruler (from the point of rotation):
T_right = r_right * F_right
Since the ruler is balanced, T_left = T_right.
Let's calculate the torques on both sides of the ruler.
On the left side:
r_left = 60.2 cm - 41.3 cm = 18.9 cm = 0.189 m
F_left = 0 (as there is no force on the left side)
T_left = r_left * F_left = 0
On the right side:
r_right = 60.2 cm - 26.2 cm = 34 cm = 0.34 m
F_right = mass * g
Here, g is the acceleration due to gravity, which is approximately 9.81 m/s^2.
To convert mass from grams to kilograms, we divide by 1000.
mass = 364 g / 1000 = 0.364 kg
F_right = 0.364 kg * 9.81 m/s^2 = 3.57344 N
T_right = r_right * F_right
= 0.34 m * 3.57344 N
= 1.215 m*N
Since T_left = T_right, we know that T_left = 1.215 m*N.
Now, let's calculate the torque of the ruler itself.
The torque due to the ruler's own weight is given by:
T_ruler = r_ruler * F_ruler
Since the ruler is balanced at the 60.2 cm mark, the distance from the 60.2 cm mark to the midpoint of the ruler is half the length of the ruler:
r_ruler = 60.2 cm / 2 = 30.1 cm = 0.301 m
F_ruler is the weight of the ruler, which can be calculated using the mass and gravitational acceleration:
mass_ruler = T_ruler / r_ruler = 1.215 m*N / 0.301 m = 4.04 N
To convert the weight from newtons to kilograms:
weight_ruler = mass_ruler * g = 4.04 N * 1 kg / 9.81 m/s^2 = 0.412 kg
Thus, the mass of the ruler is approximately 0.412 kg.