Askiitians Tutor Team
Last Activity: 4 Months ago
To find the moment of inertia of a dumb-bell system consisting of two particles with masses m1 and m2 connected by a light rod of length r, we need to understand how moment of inertia is calculated for a system of particles. The moment of inertia (I) about an axis is a measure of how difficult it is to change the rotational motion of an object around that axis. For our dumb-bell, we will calculate the moment of inertia about an axis that is perpendicular to the rod and passes through the center of mass of the system.
Understanding the Center of Mass
The center of mass (CM) of the dumb-bell can be found using the formula:
- CM = (m1 * x1 + m2 * x2) / (m1 + m2)
Assuming the rod is aligned along the x-axis, we can place m1 at position x1 = -r/2 and m2 at position x2 = r/2. Therefore, the center of mass can be calculated as:
- CM = (m1 * (-r/2) + m2 * (r/2)) / (m1 + m2)
- CM = (-(m1 * r) / 2 + (m2 * r) / 2) / (m1 + m2)
- CM = (r/2 * (m2 - m1)) / (m1 + m2)
Calculating the Moment of Inertia
The moment of inertia for a system of particles is given by the sum of the products of each mass and the square of its distance from the axis of rotation:
- I = m1 * d1^2 + m2 * d2^2
Here, d1 and d2 are the distances from the center of mass to each mass. Since we have already established the positions of m1 and m2, we can find these distances:
- d1 = distance from CM to m1 = CM - (-r/2) = (r/2 * (m2 - m1)) / (m1 + m2) + r/2
- d2 = distance from CM to m2 = CM - (r/2) = (r/2 * (m2 - m1)) / (m1 + m2) - r/2
Substituting the Distances
Now, substituting these distances into the moment of inertia formula:
- I = m1 * [(r/2 * (m2 - m1)) / (m1 + m2) + r/2]^2 + m2 * [(r/2 * (m2 - m1)) / (m1 + m2) - r/2]^2
Final Expression
After simplification, we can derive that:
- I = (m1 * m2 * r^2) / (m1 + m2)
This formula shows that the moment of inertia of the dumb-bell about the specified axis depends on both masses and the distance between them. The result illustrates how the distribution of mass relative to the axis of rotation affects the rotational inertia of the system.
In summary, the moment of inertia of the dumb-bell about an axis perpendicular to the rod passing through the center of mass is given by:
- I = (m1 * m2 * r^2) / (m1 + m2)
This result is crucial in understanding the dynamics of rotational motion in systems involving multiple masses.
