7/12/2023 0 Comments Calculate moment of inertia![]() ![]() Summer Olympics, here he comes! Confirmation of these numbers is left as an exercise for the reader. The father would end up running at about 50 km/h in the first case. In terms of revolutions per second, these angular velocities are 2.12 rev/s and 1.41 rev/s, respectively. If, for example, the father kept pushing perpendicularly for 2.00 s, he would give the merry-go-round an angular velocity of 13.3 rad/s when it is empty but only 8.89 rad/s when the child is on it. The angular accelerations found are quite large, partly due to the fact that friction was considered to be negligible. For a point mass, the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I mr 2. The angular acceleration is less when the child is on the merry-go-round than when the merry-go-round is empty, as expected. Substituting known values into the equation for gives To justify this sum to yourself, examine the definition of : ![]() The total moment of inertia is the sum of moments of inertia of the merry-go-round and the child (about the same axis). To find the total moment of inertia, we first find the child's moment of inertia by considering the child to be equivalent to a point mass at a distance of from the axis. We expect the angular acceleration for the system to be less in this part, because the moment of inertia is greater when the child is on the merry-go-round. Now, after we substitute the known values, we find the angular acceleration to be The moment of inertia of a solid disk about this axis is given in Figure to be To find the torque, we note that the applied force is perpendicular to the radius and friction is negligible, so that Note: If x L/2, we get mL³/ (24L) + mL³/ (24L) mL²/12 as seen in the video. The parallel axis thereom is used to seperate the shape into a number of simpler shapes. The moment of inertia would be mx³/ (3L) + m (L-x)³/ (3L). To solve for, we must first calculate the torque (which is the same in both cases) and moment of inertia (which is greater in the second case). The moment of inertia can be calculated by hand for the most common shapes: Rectangle: (bh3)/12 >Circle: (pi r4)/4 Triangle: (bh3)/12 If the shape is more complex then the moment of inertia can be calculated using the parallel axis thereom. StrategyĪngular acceleration is given directly by the expression : Consider the merry-go-round itself to be a uniform disk with negligible retarding friction.įigure 10.13 A father pushes a playground merry-go-round at its edge and perpendicular to its radius to achieve maximum torque. One can find the MI relative the axis z as a sum of two MI of MP being. He exerts a force of 250 N at the edge of the 50.0-kg merry-go-round, which has a 1.50 m radius.Ĭalculate the angular acceleration produced (a) when no one is on the merry-go-round and (b) when an 18.0-kg child sits 1.25 m away from the center. The required MI is Iz,C Iz ma2, where a is a distance between two parallel z axes. Example 10.7 Calculating the Effect of Mass Distribution on a Merry-Go-RoundĬonsider the father pushing a playground merry-go-round in Figure 10.13.
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