Final answer:
To stop a rolling solid sphere, the work required can be calculated by considering its kinetic energy. The sphere has both translational and rotational kinetic energy. By using the equations for kinetic energy and the relationship between linear and angular velocity, we can determine the work required. In this case, the work is approximately 720 J.
Step-by-step explanation:
To calculate the work required to stop the rolling solid sphere, we need to consider the kinetic energy of the object. As the sphere is rolling, it has both translational and rotational kinetic energy. The work done on the sphere will be equal to the change in its kinetic energy.
The kinetic energy of the sphere is given by:
KE = 0.5 * I * ω^2 + 0.5 * mv^2
Where I is the moment of inertia of the sphere, ω is its angular velocity, m is its mass, and v is its linear velocity.
Since the sphere is rolling without slipping, its angular velocity is related to its linear velocity as:
ω = v / r
Where r is the radius of the sphere.
Substituting this into the expression for kinetic energy, we get:
KE = 0.5 * (2/5 * mr^2) * (v/r)^2 + 0.5 * mv^2
Simplifying, we obtain:
KE = (7/10) * mv^2
The work done to stop the sphere is equal to the change in kinetic energy, which is the initial kinetic energy minus the final kinetic energy:
Work = KE_initial - KE_final
Since the sphere comes to a stop, its final kinetic energy is zero. Therefore, the work required to stop the sphere is:
Work = (7/10) * mv^2
Substituting the given values, we have:
Work = (7/10) * (40.0 kg) * (6.0 m/s)^2
Work ≈ 720 J