Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

As an experienced tutor registered on UrbanPro, let me guide you through this question. When it comes to analyzing rotational motion, we need to consider the principles of angular momentum and rotational inertia. In this scenario, torques of equal magnitude are applied to both a hollow cylinder and a solid sphere, each with the same mass and radius. The cylinder rotates about its standard axis of symmetry, while the sphere rotates about an axis passing through its center. To determine which object will acquire a greater angular speed after a given time, let's delve into their rotational properties: Hollow Cylinder: The rotational inertia (moment of inertia) of a hollow cylinder about its axis of symmetry is given by I=12MR2I=21MR2, where MM is the mass and RR is the radius. Since the cylinder is free to rotate about its axis of symmetry, it can be treated as a solid cylinder for this analysis. The torque applied will result in an angular acceleration, increasing its angular speed over time. Solid Sphere: The rotational inertia of a solid sphere about an axis passing through its center is given by I=25MR2I=52MR2. Similar to the cylinder, the torque applied will lead to an angular acceleration, increasing its angular speed. Comparing the rotational inertias, we can see that the solid sphere has a higher rotational inertia compared to the hollow cylinder, given the same mass and radius. Now, according to the conservation of angular momentum, the angular momentum of each object will increase as the torque is applied. However, since the solid sphere has a higher rotational inertia, it will resist changes in its rotational motion more than the hollow cylinder. As a result, the solid sphere will acquire a lower angular speed compared to the hollow cylinder after the same amount of time under the influence of the same torque. Therefore, the hollow cylinder will acquire a greater angular speed after a given time compared to the solid sphere when equal torques are applied to both. read less