Question

As part of a carnival game, a mb = 0.653kg ball is thrown at a stack of 21.3cm tall, mo = 0.343kg objects and hits with a perfectly horizontal velocity vb,i = 11.1m/s. suppose that the ball strikes the topmost object. immediately after the collision, the ball has a horizontal velocity of vb,f = 4.35m/s in the same direction, the topmost object has an angular velocity of wo = 4.83rad/s about its center of mass, and all the remaining objects are undisturbed. assume that the ball is not rotating and that the effect of torque due to gravity during the collision is negligible. if the object's center of mass is located r = 14.9 cm below the point where the ball hits, what is the moment of inertia io of the object about its center of mass? what is the center of mass velocity vo,cm of the tall object immediately after it is struck?

Answer 1

The calculation of the moment of inertia I_o involves using the principle of conservation of angular momentum, while the calculation of the center of mass velocity v_o,cm utilizes the principle of conservation of linear momentum. They both are calculated by relating initial and final states of the system following the collision.

Step-by-step explanation:

The question asks us to calculate two physics properties immediately after a perfectly horizontal collision: the moment of inertia of the object about its center of mass, and the center of mass velocity of the tall object.

Moment of Inertia Calculation

To calculate the moment of inertia Io of the object about its center of mass, we use the conservation of angular momentum principle. The initial angular momentum of the system is the product of the ball's mass mb, its initial velocity vb,i, and the distance r from the point of impact to the object's center of mass. The final angular momentum is the sum of the angular momentum of the ball, given by its mass mb, its final velocity vb,f, and the distance r, and the object's angular momentum, given by its moment of inertia Io and angular velocity wo. Equating the initial and final angular momentum and solving for Io provides its value.

Center of Mass Velocity Calculation

Similarly, the center of mass velocity vo,cm of the tall object can be calculated using the principle of conservation of linear momentum. The initial linear momentum is the product of the ball's mass and its initial velocity, and the final linear momentum includes the product of the ball's mass and its final velocity, as well as the product of the object's mass and its center of mass velocity. By subtracting the final momentum of the ball from the initial momentum and dividing by the object's mass, we get the value for vo,cm.