Final answer:
To determine the compression of a spring by a rolling ball, you must use the conservation of energy principle, relating the ball's kinetic energy to the potential energy of the spring. By equating the initial kinetic energy (translational and rotational) to the potential energy stored in the spring, you can solve for the spring's compression distance.
Step-by-step explanation:
When a ball of mass m and radius r rolls into a spring with spring constant k, the ball compresses the spring by an amount that can be found using the conservation of energy principle. The initial kinetic energy of the ball due to its motion is converted into potential energy stored in the spring as it is compressed.
To calculate the compression distance x of the spring, we equate the initial kinetic energy of the ball (which is the sum of translational and rotational kinetic energy if the ball is rolling without slipping) to the potential energy stored in the spring (1/2*k*x2). The equation to solve for x is: 1/2*m*v2 + 1/2*I*ω2 = 1/2*k*x2, where I is the moment of inertia of the ball and ω is the angular velocity.
For a solid sphere (like a ball), the moment of inertia I is (2/5)*m*r2, and the angular velocity ω can be related to the translational velocity v by the relation ω = v/r. Substituting these into the equation and solving for x will provide the compression of the spring in terms of the given variables m, r, v, and k.