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ROTATION AND RIGID BODIES - Coggle Diagram
ROTATION AND RIGID BODIES
Angular Quantities
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined:
where l is the arc length.
Angular displacement:
The average angular velocity is defined as the total angular displacement divided by time:
The instantaneous angular velocity:
The angular acceleration is the rate at which the angular velocity changes with time:
The instantaneous acceleration:
Every point on a rotating body has an angular
velocity ω and a linear velocity v.
Vector Nature of Angular Quantities
The angular velocity vector points along the axis of rotation, with the direction given by the right- hand rule. If the direction of the rotation axis does not change, the angular acceleration vector points along it as well.
Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.
Torque
To make an object start rotating, a force is needed; the position and direction of the force matter as well.
The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.
A longer lever arm is very helpful in rotating objects.
Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.
Rotational Dynamics; Torque and Rotational Inertia
This is for a single point mass; what about an extended object?
As the angular acceleration is the same for the whole object, we can write:
is called the rotational inertia of an object.
The distribution of mass matters here—these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.
The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation—compare (f) and (g), for example.
Solving Problems in Rotational Dynamics
Draw a diagram.
Decide what the system comprises.
Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act.
Find the axis of rotation; calculate the torques around it.
Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step.
Apply Newton’s second law for translation and other laws and principles as needed.
Solve.
Check your answer for units and correct
order of magnitude.
Determining Moments of Inertia
If a physical object is available, the moment of inertia can be measured experimentally.
Otherwise, if the object can be considered to be a continuous distribution of mass, the moment of inertia may be calculated:
The parallel-axis theorem gives the moment of inertia about any axis parallel to an axis that goes through the center of mass of an object:
The perpendicular-axis theorem is valid only for flat objects.
Rotational Kinetic Energy
The kinetic energy of a rotating object is given by
By substituting the rotational quantities, we find that the rotational kinetic energy can be written:
A object that both translational and rotational motion also has both translational and rotational kinetic energy:
When using conservation of energy, both rotational and translational kinetic energy must be taken into account.
All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational
inertia they have.
The torque does work as it moves the wheel
through an angle θ:
