A solid sphere of mass m and radius R is rolling without slipping on a horizontal surface with a translational speed v . It then encounters an upward inclined plane. In the first case, the incline is rough enough to support pure rolling, and the sphere reaches a maximum vertical height h₁ . In the second case, the incline is perfectly smooth, and the sphere reaches a maximum vertical height h₂ . If the ratio h₁ h₂ is

Step-by-step solution

For a solid sphere, the moment of inertia is I = 2 5 mR^2 , which means k^2 R^2 = 2 5 . In the first case (rough incline), the sphere rolls without slipping until its translational and rotational velocities both become zero. The total kinetic energy is converted into potential energy: mgh₁ = 1 2 mv^2 (1 + k^2 R^2 ) = 1 2 mv^2 (1 + 2 5 ) = 7 10 mv^2 h₁ = 7v^2 10g In the second case (perfectly smooth incline), there is no friction on the incline to provide a torque. Thus, the angular velocity of the sphere remains co