Two spheres of mass 1.0kg and 4.0kg are joined by a light rod so that their centres are 0.20m apart. Calculate the position of centre of gravity
mass1 * (distance1 from CG) = mass2 * (distance2 from CG)
distance1 + distance2 = 0.20 m
The position of the center of gravity can be calculated by taking the mass-weighted average of the individual positions of the spheres. The equation for calculating the position of the center of gravity is:
x = (m1 * x1 + m2 * x2) / (m1 + m2)
Where x represents the position of the center of gravity, m represents mass, and the subscripts 1 and 2 correspond to the two spheres.
Given:
Mass of the first sphere (m1) = 1.0 kg
Mass of the second sphere (m2) = 4.0 kg
Distance between their centers (r) = 0.20 m
To find the positions of the spheres, we need to understand the geometry of the situation:
Let's assume the first sphere is located at position x1 and the second sphere is located at position x2. The distance between them is given as 0.20 m.
Since the two spheres are joined by a light rod, the distance between their centers (0.20 m) is equal to the sum of their individual distances from the center of gravity. So, we can express this relationship as:
x2 - x1 = r
Rearranging this equation, we get:
x2 = x1 + r
Now, substituting this value of x2 in the equation for the position of the center of gravity, we have:
x = (m1 * x1 + m2 * (x1 + r)) / (m1 + m2)
Simplifying further:
x = (m1 * x1 + m2 * x1 + m2 * r) / (m1 + m2)
x = (x1 * (m1 + m2) + m2 * r) / (m1 + m2)
x = (x1 * m1 + x1 * m2 + m2 * r) / (m1 + m2)
x = x1 * (m1 + m2) / (m1 + m2) + r * (m2 / (m1 + m2))
x = x1 + r * (m2 / (m1 + m2))
In this equation, we need to find the value of x1. Since the position of x1 is not given, we can choose any arbitrary value. For simplicity, we can take x1 as zero, which means considering the first sphere as the reference point.
Now, plugging in the values:
x = 0 + 0.20 * (4.0 / (1.0 + 4.0))
x = 0.20 * (4.0 / 5.0)
x = 0.16 m
Therefore, the position of the center of gravity is located 0.16 meters from the first sphere along the axis joining both spheres.