Rotational kinematics
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Rotational Kinematics. Rotation about a fixed axis. Rotational Motion Remember: a circle is just a straight line rolled up

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Rotational Kinematics


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Rotation about a fixed axis

  • Rotational Motion

    • Remember: a circle is just a straight line rolled up

    • If you spin a wheel, and look at how fast a point on the wheel is spinning, the answer depends on how far away the point is from the center. Linear velocity isn't the most convenient quantity to use when you're dealing with rotation, and for the same reason neither is displacement, or acceleration;

    • it is often more convenient to use rotational equivalents.

    • The equivalent variables for rotation are:

      • angular displacement (angle, for short)

      • angular velocity

      • angular acceleration

      • All the angular variables are related to the straight-line variables by a factor of r, the distance from the center of rotation to the point you're interested in.

I think my head is spinning


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Rotation about a fixed axis

  • Rolling Motion

    • When an object such as a wheel or a ball rolls, it does not slip where it makes contact with the ground.

    • With a car or a bicycle tire, there is friction between the tire and the road, and if the tire is rolling then the frictional force is a static force of friction.

      • This is because there is no slipping, so the point on the tire in contact with the road is instantaneously at rest.

    • For a point on the outside of the tire, the rotational speed happens to be equal to the linear speed of the car: this is because each time the tire makes a complete revolution, the car will have traveled a distance equal to the circumference of the tire, so the linear distance and the rotational distance are the same for the same time interval.

Now we’re rolling!


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Rotational Dynamics


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Rotational Dynamics

  • Torque

    • A torque is a force exerted at a distance from an axis of rotation; the easiest way to think of torque is to consider a door.

      • When you open a door, where do you push?

        • If you exert a force at the hinge, the door will not move

          • the easiest way to open a door is to exert a force on the side of the door opposite the hinge, and to push or pull with a force perpendicular to the door. This maximizes the torque you exert.

    • We can state the equation for torque as


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Rotational Dynamics

  • Center of Gravity

    • The center of gravity of an object is the point you can suspend the object from without there being any rotation because of the force of gravity, no matter how the object is oriented. If you suspend an object from any point, let it go and allow it to come to rest, the center of gravity will lie along a vertical line that passes through the point of suspension. Unless you've been exceedingly careful in balancing the object, the center of gravity will generally lie below the suspension point.

    • The center of gravity is an important point to know, because when you're solving problems involving large objects, or unusually-shaped objects, the weight can be considered to act at the center of gravity. In other words, for many purposes you can assume that object is a point with all its weight concentrated at one point, the center of gravity.

    • The center of mass of an object is generally the same as its center of gravity. Very large objects, large enough that the acceleration due to gravity varies in different parts of the object, are the only ones where the center of mass and center of gravity are in different places.

      • An object thrown through the air may spin and rotate, but its center of gravity will follow a smooth parabolic path, just like a ball.


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Rotational Dynamics

  • Moment of Inertia (Rotational Mass)

    • the rotational equivalent of mass, which is something called the moment of inertia.

    • Mass is a measure of how difficult it is to get something to move in a straight line, or to change an object's straight-line motion. The more mass something has, the harder it is to start it moving, or to stop it once it starts. Similarly, the moment of inertia of an object is a measure of how difficult it is to start it spinning, or to alter an object's spinning motion. The moment of inertia depends on the mass of an object, but it also depends on how that mass is distributed relative to the axis of rotation: an object where the mass is concentrated close to the axis of rotation is easier to spin than an object of identical mass with the mass concentrated far from the axis of rotation.

    • The moment of inertia of an object depends on where the axis of rotation is. The moment of inertia can be found by breaking up the object into little pieces, multiplying the mass of each little piece by the square of the distance it is from the axis of rotation, and adding all these products up:

      • For common objects rotating about typical axes of rotation, these sums have been worked out, so we don't have to do it ourselves. A table of some of these moments of inertia can be found on the next slide

Divers minimizing their moments of inertia in order to increase their rates of rotation.


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Rotational Dynamics

  • Moment of Inertia


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Rotational Dynamics

  • Angular Momentum

    • The angular momentum of a moving particle is

    • where m is its mass, v⊥ is the component of its velocity vector perpendicular to the line joining it to the axis of rotation, and r is its distance from the axis. Positive and negative signs are used to describe opposite directions of rotation.

    • The angular momentum of a finite-sized object or a system of many objects is found by dividing it up into many small parts, applying the equation to each part, and adding to find the total amount of angular momentum.

    • Conservation of angular momentum has been verified over and over again by experiment, and is now believed to be one of the three most fundamental principles of physics, along with conservation of energy and momentum.


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Rotational Dynamics

  • Angular Momentum Conserved

    • When a figure skater is twirling, there is very little friction between her and the ice, so she is essentially a closed system, and her angular momentum is conserved. If she pulls her arms in, she is decreasing r for all the atoms in her arms. It would violate conservation of angular momentum if she then continued rotating at the same speed, i.e., taking the same amount of time for each revolution, because her arms' contributions to her angular momentum would have decreased, and no other part of her would have increased its angular momentum. This is impossible because it would violate conservation of angular momentum. If her total angular momentum is to remain constant, the decrease in r for her arms must be compensated for by an overall increase in her rate of rotation. That is, by pulling her arms in, she substantially reduces the time for each rotation.


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Rotational Dynamics

  • Rotational and Translational Analogies


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Rotational Dynamics

  • Application of Conservation of Angular Momentum and Torque

    • Angular displacement, angular velocity, and angular acceleration are all vectors, too. But which way do they point? Every point on a rolling tire has the same angular velocity, and the only way to ensure that the direction of the angular velocity is the same for every point is to make the direction of the angular velocity perpendicular to the plane of the tire.

      • To figure out which way it points, use your right hand.

        • Stick your thumb out as if you're hitch-hiking, and curl your fingers in the direction of rotation. Your thumb points in the direction of the angular velocity.

    • If you look directly at something and it's spinning clockwise, the angular velocity is in the direction you're looking; if it goes counter-clockwise, the angular velocity points towards you. Apply the same thinking to angular displacements and angular accelerations.


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Rotational Dynamics

  • Application of Conservation of Angular Momentum and Torque

    • Gyroscopes

      • A gyroscope is a device for measuring or maintaining orientation, based on the principle of conservation of angular momentum. The essence of the device is a spinning wheel on an axle.

      • The classic image of a gyroscope is a fairly massive rotor suspended in light supporting rings called gimbals which have nearly frictionless bearings and which isolate the central rotor from outside torques. At high speeds, the gyroscope exhibits extraordinary stability of balance and maintains the direction of the high speed rotation axis of its central rotor. The implication of the conservation of angular momentum is that the angular momentum of the rotor maintains not only its magnitude, but also its direction in space in the absence of external torque. The classic type gyroscope finds application in gyro-compasses, but there are many more common examples of gyroscopic motion and stability. Spinning tops, the wheels of bicycles and motorcycles, the spin of the Earth in space, even the behavior of a boomerang are examples of gyroscopic motion.

Gyroscopes are used in inertial navigation systems to navigate jetliners and ships across the oceans