In a vertical plane, a closed circular tube of radius 10 cm contains a block of mass 100 g that just fits inside its cross-section. The block is propelled with a speed of 5.0 m/s at the highest point of the tube. It executes multiple oscillations inside the tube before finally coming to rest at the lowest point. Determine the work done by the tube on the block during this process.

Step-by-step solution

Let the lowest point of the tube be the reference level for gravitational potential energy ( h = 0 ). At the highest point, the height of the block is h = 2R = 2 0.1 = 0.2 m . The initial mechanical energy of the block is the sum of its kinetic and potential energies: E_i = K_i + U_i = 1 2 mv^2 + mgh Substituting the given values ( m = 0.1 kg , v = 5.0 m/s , g = 10 m/s ^2 ): E_i = 1 2 (0.1)(5.0)^2 + (0.1)(10)(0.2) E_i = 1.25 + 0.20 = 1.45 J The block finally comes to rest at the lowest point, so its final velocity