Final answer:
The maximum height attained by a rolling disk can be found using the conservation of energy, equating the disk's initial kinetic energy (both translational and rotational) to its gravitational potential energy at the maximum height.
Step-by-step explanation:
To find the maximum height h a rolling disk will attain, we use the principle of conservation of energy. The initial kinetic energy of the disk is partly translational (due to its velocity) and partly rotational (due to its angular velocity).
As the disk rolls up the surface, this kinetic energy is converted into gravitational potential energy. At the maximum height, all the kinetic energy will be converted into potential energy, assuming there is no energy loss.
To calculate the height, we equate the initial kinetic energy to the potential energy at the maximum height.
The formula for the initial kinetic energy (KE) of a rolling object is given by KE = (1/2)mv² + (1/2)Iω², where m is the mass of the object, v is the velocity, I is the moment of inertia, and ω is the angular velocity. For a disk rolling without slipping, ω = v/R, where R is the radius of the disk.
The potential energy (PE) at height h is given by PE = mgh, where g is the acceleration due to gravity (9.8 m/s²). Setting the kinetic energy equal to the potential energy and solving for h gives us the maximum height reached by the disk.
The moment of inertia for a solid disk is (1/2)mR², leading to the initial kinetic energy formula KE = (1/2)mv² + (1/4)mv² = (3/4)mv². Equating this to mgh allows us to solve for h as h = (3/4)v²/g.