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10.3 Dynamics of Rotational Motion: Rotational Inertia

- A person uses a microwave oven to cook lunch.
- A fly lands on the outer edge of the rotating plate after accidentally flying into the microwave.
- The total distance traveled by the fly during a 2.0-min cooking period is calculated if the plate has a radius of 0.15 m.

- First, find the total number of revolutions and then the distance traveled.

- The total distance traveled by the fly is shown here.
- For complete revolutions, displacement is zero because they bring the fly back to its original position.
- The first mention of the distinction between total distance traveled and displacement was in One-Dimensional Kinematics.

- Many useful relationships are expressed in equation form.

- The laws of nature are not represented by rotational kinematics.
- We can describe many things to great precision, but we don't consider causes.
- A very rapid change in angular velocity is described by a large angular acceleration.

- Many of the factors that are involved are predicted by your intuition.
- If we push too close to the hinges, a door will open slowly.
- The more massive the door, the slower it opens.
- One implication is that the greater the force applied from the pivot, the greater the angular acceleration.
- The familiar relationships among force, mass, and acceleration embodied in the second law of motion should be familiar to these relationships.
- There are precise rotational analogs to both force and mass.

- The bike wheel needs force to spin.
- The greater the force, the faster the acceleration.
- The angular acceleration will be smaller if you push on a spoke closer to the axle.

- An acceleration can be obtained in the direction of the force.
- We can rearrange this equation so that we can relate it to expressions for rotational quantities.

- Torque is simply because it is perpendicular to.

- If we add both sides of the equation together, we get Torque on the left-hand side.

- The last equation is an approximation ofNewton's second law, which states that Torque is analogous to force, angular acceleration is analogous to translation, and inertia is analogous to mass.

- An object is supported by a horizontal table and attached to a pivot point by a cord.
- The force applied to the object causes it to accelerate.
- The force is constant.

- Linear or translational dynamics are completely analogous to the dynamics of rotational motion.
- Dynamics is concerned with mass and force.
- We will find direct analogs to force and mass that behave the same as they did before.

- This is similar to in motion.
- The moment of inertia for any object depends on the axis.
- calculating is beyond the scope of this text except for one simple case--that of a hoop, which has all its mass at the same distance from its axis.
- A hoop's inertia around its axis is where its total mass and radius are.

- The table is a piece of artwork that has shapes as well as formulae, so we must consult Figure 10.12 for formulas that have been derived from integration over the continuous body.
- We might expect units of mass to be divided by distance squared.

- We will only consider the forces in the plane of rotation.
- Torques are either positive or negative and add like ordinary numbers.
- The relationship in is very similar to the second law.
- The equation is valid for any Torque, relative to any axis.

- The larger the Torque is, the bigger the angular acceleration is.
- The quicker a child pushes on a merry-go-round, the faster it will accelerate.
- The faster the merry-go-round is, the slower it is.
- The larger the moment of inertia, the smaller the acceleration.
- There is more than one twist.
- The moment of inertia is dependent on the mass of the object, as well as its distribution of mass relative to the axis around which it rotates.
- It will be easier to accelerate a merry-go-round full of children if they stand close to its axis.
- The mass is the same, but the moment of inertia is larger when the children are at the edge.

- Cut out a circle from cardboard.
- The circle should be positioned so that it can move freely through the center of the horizontal axis.

- Torque and mass are involved in the rotation.
- Draw a picture of the situation.

- The system of interest should be determined.

- A body diagram can be drawn.
- Draw and label the external forces that are acting on the system of interest.

- To solve the problem, apply the rotational equivalent ofNewton's second law.
- Care must be taken to use the correct moment of inertia and to consider the point of rotation.

- Check the solution to see if it's reasonable.

- The net Torque is zero in statics.
- In the second law of motion for rotation, net Torque is the cause of acceleration.

- A father pushes a merry-go-round at the edge of the playground.

- The moment of inertia in the second case is greater than in the first case, so we must first calculate the Torque and Moment of inertia.

- The moment of inertia is greater when the child is on the merry-go-round, so we expect the system's acceleration to be less in this part.
- The child's moment of inertia is equivalent to a point mass at a distance of 1.25 m from the axis.

- The inertia of the merry-go-round and the child is the total moment of inertia.

- When the merry-go-round is empty, the child's acceleration is less than expected.

- The large angular accelerations were found due to the fact that there was no friction.

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