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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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