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# UAE Inspire Science Physics, Student Edition, 2024-25 ## PRACTICE Problems 1. Two children of equal masses sit 0.3 m from the center of a seesaw. Assuming that their masses are much greater than that of the seesaw, by how much is the moment of inertia increased when they sit 0.6 m from the cent...

# UAE Inspire Science Physics, Student Edition, 2024-25 ## PRACTICE Problems 1. Two children of equal masses sit 0.3 m from the center of a seesaw. Assuming that their masses are much greater than that of the seesaw, by how much is the moment of inertia increased when they sit 0.6 m from the center? Ignore the moment of inertia for the seesaw. 2. Suppose there are two balls with equal diameters and masses. One is solid and the other is hollow, with all its mass distributed at its surface. Are the moments of inertia of the balls equal? If not, which is greater? 3. Calculate the moments of inertia for each object below using the formulas in Table 2. Each object has a radius of 2.0 m and a mass of 1.0 kg. * a thin hoop * a solid, uniform cylinder * a solid, uniform sphere ## ADDITIONAL PRACTICE 24. **CHALLENGE** Figure 13 shows three equal-mass spheres on a rod of very small mass. Consider the moment of inertia of the system, first when it is rotated about sphere A and then when it is rotated about sphere C. * Are the moments of inertia the same of different? Explain. If the moments of inertia are different, in which case is the moment of inertia greater? * Each sphere has a mass of 0.10 kg. The distance between spheres A and C is 0.20 m. Find the moment of inertia in the following instances: rotation about sphere A, rotation about sphere C. ## Newton's Second Law for Rotational Motion Newton's second law for linear motion is expressed as $a = \frac{F_{net}}{m}$. If you rewrite this equation to represent rotational motion, acceleration is replaced by angular acceleration $(\alpha)$, force is replaced by net torque $(\tau_{net})$, and mass is replaced by moment of inertia (I). Angular acceleration is directly proportional to the net torque and inversely proportional to the moment of inertia as stated in Newton's second law for rotational motion. This law is expressed by the following equation. ### Newton's Second Law for Rotational Motion The angular acceleration of an object about a particular axis equals the net torque on the object divided by the moment of inertia. $\alpha = \frac{\tau_{net}}{I}$ If the torque on an object and the angular velocity of that object are in the same direction, then the angular velocity of the object increases. If the torque and angular velocity are in different directions, then the angular velocity decreases. ## PHYSICS Challenge ### Moments of Inertia Rank the objects shown in the diagram from least to greatest according to their moments of inertia about the indicated axes. All spheres have equal masses and all separations are the same. Assume that the rod's mass is negligible. The image shows four objects: * Object A: A single sphere * Object B: Two spheres * Object C: Three spheres * Object D: Four spheres Each object is connected by a rod with the spheres lined up vertically.