Question

A uniform solid phere of radius 2.00cm and mass 4.50kg starts with a purely translational inclined plane. the surface of the incline is 1.25m long, and is tilted at an angle of 30 with respect to the horizon. assuming the sphere rolls without slipping down the incline, calculate the sphere's final translational speed

Answer 1

To find the sphere's final translational speed, we use conservation of energy, including both translational and rotational kinetic energy, and relate them to the potential energy at the top of the incline.

Step-by-step explanation:

The problem can be solved using the principle of conservation of energy. The potential energy of the sphere at the top of the inclined plane is converted to translational kinetic energy and rotational kinetic energy as it rolls down. The potential energy (PE) at the top is given by mgh, where m is the mass of the sphere, g is the acceleration due to gravity, and h is the height of the incline. This potential energy equals the sum of translational kinetic energy (1/2mv²) and rotational kinetic energy (1/2Iω²), where I is the moment of inertia of the sphere and ω is the angular velocity.

For a solid sphere, the moment of inertia is (2/5)mR² and because the sphere rolls without slipping, the angular velocity ω is related to the translational velocity v by the equation ω = v/R. We can calculate the height h using trigonometry knowing the length of the incline (1.25m) and the angle of the incline (30 degrees). Finally, we solve for v using these relationships.