System Of Particles Rotational Motion

Step 1: Total Energy at the Bottom

At the bottom of the ring:

Translational Kinetic Energy:

Rotational Kinetic Energy:

For a rolling object without slipping, .

For a solid sphere, moment of inertia .

Total Kinetic Energy:

Potential Energy: Considered zero at the bottom.

Step 2: Total Energy at the Top

At the top of the ring:

Height Change:

Potential Energy:

Kinetic Energy (with velocity ):

Step 3: Conservation of Energy

Total energy at bottom = Total energy at top:

Mass cancels out:

Step 4: Substitute Values and Solve

Given:

Given:

• A wheel is rolling on a plane surface.
• A point on the rim (at the same height as the center) has speed .

Step 1: Understand Rolling Motion

For a rolling wheel:

• The center moves with speed .
• Due to rotation, points on the rim have additional tangential speed , where is the radius and is angular velocity.
• In pure rolling, .

Step 2: Analyze the Given Point

At the same height as the center, the point's total velocity is the vector sum of:

1. Translational velocity of the center: (horizontal).
2. Tangential velocity due to rotation: (horizontal, opposite to translation).

Thus, the net speed at this point is:

(The negative sign accounts for the opposite direction of rotation at this point.)

Step 3: Solve for

Given :

Conclusion:

Based on standard rolling motion analysis, the center speed should be 2 m/s.

Moment of Inertia of a Single Sphere:

• The moment of inertia of a solid sphere about its center of mass is , where M is the mass and R is the radius.
• We need the moment of inertia about a tangent to the sphere. We'll use the parallel axis theorem:
  • (distance from the center of mass to the tangent)
  • Therefore,

1. Moment of Inertia of the System:
   • Since there are two identical spheres, the total moment of inertia is:
2. Solve for Mass (M):
   • We're given that and . Plug these values into the equation:
   • Solve for M:

Given:

• Mass of the rod, (uniform, length 1 m)
• Point mass, , attached at 40 cm from the rod's center

Step 1: Define Coordinate System

Place the rod along the x-axis with its original center of mass (COM) at .

The point mass is attached at .

Step 2: Calculate New COM Position

The new COM ( ) is given by:

Step 3: Determine Shift in COM

The shift in COM is equal to the new COM position since the original COM was at 0:

Conclusion:

The center of mass shifts by 10 cm.

Answer: