three 80 gram weights are suspended at 32 cm 43 cm and 76 cm of a uniform meter stick. if the stick is made to rotate at its midpoint where must the 4th 80 gram weight be suspended to attain equilibrium? please and thank you
To determine where the 4th 80-gram weight must be suspended to attain equilibrium, we need to apply the principle of moments. The principle of moments states that for an object in rotational equilibrium, the sum of the clockwise moments (torque) about any point must equal the sum of the counterclockwise moments (torque) about the same point.
In this case, the uniform meter stick rotates at its midpoint, so we can consider this point as the fulcrum or axis of rotation. Let's name the distances of the three already suspended weights from this midpoint:
Weight 1: 32 cm
Weight 2: 43 cm
Weight 3: 76 cm
The weights can be considered as point masses, so their torques can be calculated by multiplying the weight of each mass by its distance from the midpoint. Since the weights are all 80 grams (or 0.08 kg) and we are interested in achieving equilibrium, the sum of the clockwise torques must equal the sum of the counterclockwise torques.
Let's solve for the unknown distance, which we'll call x:
(32 cm)(0.08 kg) + (43 cm)(0.08 kg) + (76 cm)(0.08 kg) = (x cm)(0.08 kg)
Now we can solve for x:
(32 cm)(0.08 kg) + (43 cm)(0.08 kg) + (76 cm)(0.08 kg) = (x cm)(0.08 kg)
2.56 kg·cm + 3.44 kg·cm + 6.08 kg·cm = (x cm)(0.08 kg)
12.08 kg·cm = (x cm)(0.08 kg)
Dividing both sides by 0.08 kg, we get:
x cm = 12.08 kg·cm / 0.08 kg
x cm = 151 cm
Therefore, the 4th 80-gram weight should be suspended at a distance of 151 cm from the midpoint to attain equilibrium.
To find the position where the fourth 80-gram weight must be suspended for equilibrium, we need to consider the principle of moments. The principle of moments states that the sum of the clockwise moments is equal to the sum of the anticlockwise moments.
First, we need to calculate the moments of the first three 80-gram weights about the midpoint of the meter stick.
Moment of the first weight = Weight × Distance from the midpoint
Moment of the first weight = 80 g × 32 cm (0.32 m)
Moment of the second weight = 80 g × 43 cm (0.43 m)
Moment of the third weight = 80 g × 76 cm (0.76 m)
Now, let's assume the fourth 80-gram weight is suspended at a distance "x" from the midpoint. The equation for equilibrium can be set up as follows:
Sum of clockwise moments = Sum of anticlockwise moments
(0.32 m × 80 g) + (0.43 m × 80 g) + (0.76 m × 80 g) = (x m × 80 g)
Simplifying the equation, we have:
25.6 + 34.4 + 60.8 = 80x
120.8 = 80x
To solve for "x," divide both sides of the equation by 80:
x = 120.8 / 80
x = 1.51 m
Therefore, the fourth 80-gram weight must be suspended at 1.51 meters from the midpoint of the meter stick in order to attain equilibrium.