Systems Of Particles And Rotational Motion

Step 1: Understanding the Concept of Center of Mass

- The spheres are made of the same material, meaning their masses are proportional to their volumes. - The mass of a sphere is given by: where is the density of the material.
Step 2: Center of Mass Formula for Two Particles

- The center of mass for two objects is given by:
Step 3: Substituting Masses of Spheres

Since mass is proportional to :
Step 4: Conclusion

Since the distance of the center of mass from the point of contact is , Option (4) is correct.

Step 1: Identify the Geometry:
We have a regular hexagon with side length m. Mass of each particle . The axis of rotation passes through two diagonally opposite vertices (e.g., A and D).
Step 2: Determine Perpendicular Distances ( ):
Moment of Inertia , where is the perpendicular distance of the -th mass from the axis.

• Vertices on Axis (2 particles): The distance . Contribution .
• Vertices off Axis (4 particles): In a regular hexagon, the distance of the vertices (say B, C, E, F) from the main diagonal (AD) is equal to the altitude of the equilateral triangle formed by the center and two adjacent vertices. Distance .

Step 3: Calculate Moment of Inertia:
Total comes from the 4 particles at distance .
Step 4: Substitute Values:
kg. m .
Step 5: Final Answer:
The moment of inertia is 90 .

1. Moment of inertia of thin disc about diameter: .
2. Masses given . Radii same (thickness same, same material).
3. So ? Wait – check: diameters – for thin discs about **diameter**, , but if **discs have same area ratio as mass**, ratio of cancels mass effect. Hence ratio is 1:1.

• Let the tensions at 20 cm and 70 cm be and respectively.
• The weight of the metre scale, N, acts at the center (50 cm mark).
• Taking moments about the 20 cm mark: • Hence, the tension at the 70 cm mark is 4.2 N.