A sphere begins rolling down an incline with an inclination . Determine the speed of its centre once it has travelled a distance l .

Step-by-step solution

By conservation of mechanical energy, the loss in potential energy equals the gain in kinetic energy. mgh = 1 2 mv^2 + 1 2 I ^2 For a solid sphere, the moment of inertia is I = 2 5 mR^2 . For pure rolling, v = R . The vertical distance fallen is h = l . Substituting these values into the energy equation: mgl = 1 2 mv^2 + 1 2 ( 2 5 mR^2 ) ( v R )^2 mgl = 1 2 mv^2 + 1 5 mv^2 mgl = 7 10 mv^2 v^2 = 10 7 gl v = 10 7 gl