An Unusual Baton Four tiny spheres are fastened to the ends of two rods of negligible mass lying in the xy-plane to form an unusual baton (see figure). We shall assume the radii of the spheres are small compared with the dimensions of the rods. Four spheres form an unusual baton. (a) The baton is rotated about the y axis. (b) The baton is rotated about the z-axis. (a) If the system rotates about the y-axis (figure (a)) with an angular speed w, find the moment inertia and the rotational kinetic energy of the system about this axis. SOLUTION Conceptualize The figure is a pictorial representation that helps conceptualize the system spheres and how it spins. Model the spheres as particles. Categorize This example is a substitution problem because it is a straightforward application of the definitions discussed in this section. (Use the following as necessary: M, a, m, b, and a.) Apply the equation for the moment of inertia to the system: - Ma? + Ma? - Evaluate the rotational kinetic energy using the following equation: That the two spheres of mass m do not enter into this result makes sense because they have no motion about the axis of rotation; hence, they have no rotational kinetic energy. By similar logic, we expect the moment of inertia about the x-axis to be I with a rotational kinetic energy about that axis of K. = mb*o?. (b) Suppose the system rotates in the xy-plane about an axis (the z-axis) through the center of the baton (figure (b)). Calculate the moment of Inertia and rotational kinetic energy about this axis. SOLUTION (Use the following as necessary: M, a, m, b, and a.) Apply the equation for the moment of inertia this new rotation axis: 2-2mr? - Ma? + Ma? + mb? + mb2 Evaluate the rotational kinetic energy using the following equation: KR - -(2Ma? + 2mb²yw² Comparing the results for parts (a) and (b), we conclude that the moment of inertia and therefore the rotational kinetic energy associated with a given angular speed -Select- indicates it would require ---Select-- Von the axis of rotation. In part (b), we expect the result to include all four spheres and distances because all four spheres are rotating in the xy-plane. Based on the work-kinetic energy theorem, the smaller rotational kinetic energy in part (a) than in part (b) Vwork to set the system into rotation about the y-axis than about the z-axis. EXERCISE Two 15.4 kg masses are now fastened to the frame on the z-axis with the two masses at z = +0.60 m. It is now given that M = 10.0 kg, m = 8.2 kg, a = 0.80 m, and b = 0.40 m. Hint (a) Find I (in kg • m3). |kg - m2 (b) Find I, (in kg - m). kg - m? Need Help? Read It

An Unusual Baton Four tiny spheres are fastened to the ends of two rods of negligible mass lying in the xy-plane to form an unusual baton (see figure). We shall assume the radii of the spheres are small compared with the dimensions of the rods. Four spheres form an unusual baton. (a) The baton is rotated about the y axis. (b) The baton is rotated about the z-axis. (a) If the system rotates about the y-axis (figure (a)) with an angular speed w, find the moment inertia and the rotational kinetic energy of the system about this axis. SOLUTION Conceptualize The figure is a pictorial representation that helps conceptualize the system spheres and how it spins. Model the spheres as particles. Categorize This example is a substitution problem because it is a straightforward application of the definitions discussed in this section. (Use the following as necessary: M, a, m, b, and a.) Apply the equation for the moment of inertia to the system: - Ma? + Ma? - Evaluate the rotational kinetic energy using the following equation: That the two spheres of mass m do not enter into this result makes sense because they have no motion about the axis of rotation; hence, they have no rotational kinetic energy. By similar logic, we expect the moment of inertia about the x-axis to be I with a rotational kinetic energy about that axis of K. = mb*o?. (b) Suppose the system rotates in the xy-plane about an axis (the z-axis) through the center of the baton (figure (b)). Calculate the moment of Inertia and rotational kinetic energy about this axis. SOLUTION (Use the following as necessary: M, a, m, b, and a.) Apply the equation for the moment of inertia this new rotation axis: 2-2mr? - Ma? + Ma? + mb? + mb2 Evaluate the rotational kinetic energy using the following equation: KR - -(2Ma? + 2mb²yw² Comparing the results for parts (a) and (b), we conclude that the moment of inertia and therefore the rotational kinetic energy associated with a given angular speed -Select- indicates it would require ---Select-- Von the axis of rotation. In part (b), we expect the result to include all four spheres and distances because all four spheres are rotating in the xy-plane. Based on the work-kinetic energy theorem, the smaller rotational kinetic energy in part (a) than in part (b) Vwork to set the system into rotation about the y-axis than about the z-axis. EXERCISE Two 15.4 kg masses are now fastened to the frame on the z-axis with the two masses at z = +0.60 m. It is now given that M = 10.0 kg, m = 8.2 kg, a = 0.80 m, and b = 0.40 m. Hint (a) Find I (in kg • m3). |kg - m2 (b) Find I, (in kg - m). kg - m? Need Help? Read It