How do we describe rotational motion?
Assume you make a chalk mark on the rim of the wheel and you originally orient the wheel so that the chalk mark is facing you. This is your reference orientation. The angular position of the wheel describes its orientation relative to this reference orientation. As the wheel rotates, this angular position is changing. The angular displacement measures how far it has rotated from its reference orientation. It is often convenient to orient a coordinate system such that the z-axis coincides with the axis of rotation and the x-axis defines the reference orientation. Then the angular displacement q of a point P on the wheel is the angle q a line from the axis of rotation to the point P makes with the x-axis.
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Angles can be measured in units of degrees or radians. 360 degrees
= 2p radians. When
describing rotational motion it is most convenient to measure angles in units of
radians. To find out how fast the wheel is rotating, we measure its angular speed w. The average angular speed is given by
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Assume that you turn the axle of the spinning wheel from vertical to horizontal. The wheel is still spinning with the same angular speed, but its angular velocity w has changed. Angular velocity is a has magnitude and direction. Its magnitude is the angular speed, and its direction is the direction of the axis of rotation. There is, however, a subtlety we have to take care of. Assume that the axis of rotation is vertical. What is the sense of rotation? Is the wheel spinning clockwise or counterclockwise as viewed from above? Just saying the axis is vertical does not tell us the sense of rotation.
To specify the sense of rotation we use a convention called the right-hand rule. If the fingers of your right hand are curling to indicate which way the wheel is turning, then the thumb of you right hand is pointing in the direction of the axis of rotation. This is the direction of the angular velocity.
| The angular acceleration a
is defined as the rate of change of the angular velocity. The average angular
acceleration is given by
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The angular velocity changes when the rotation rate is increasing or decreasing and when the axis of rotation changes direction. |
What causes angular acceleration?
| Assume you want to change a rotating wheel's angular speed. To increase
the angular speed you probably will apply a force to the rim, tangential to
the rim, and in the direction of the instantaneous velocity of the section
of the rim to which you apply the force.
If you want to decrease the angular speed, you will reverse the direction of the force. |
Assume you want to enter a building with a rotating door. The door has
four panels, and you push on one of them, perpendicular to the surface of
the panel.
The rate, at which the angular velocity of the door changes, i.e. the angular acceleration a, is greater the farther away from the axis of rotation you apply the force. |
Angular acceleration about a point is the result of a torque about this point. A torque is the product of a lever arm and a force that is applied perpendicular to the lever arm. The lever arm or moment arm is the perpendicular distance from the center of rotation, i.e. from the pivot point, to the point where the force is applied. A torque is always defined with respect to a pivot point.
A larger torque produces a larger angular acceleration. You can get a larger torque by applying a larger force, or by using a longer lever arm. We write
torque = lever arm ´ force,
A torque
t produces an angular acceleration a about a point. What is the magnitude of this angular acceleration?Newton's second law, when applied to rotational motion states that the torque equals the product of the moment of inertia I and the angular acceleration
a.torque = moment of inertia times angular acceleration
t = I * aWhen two objects are acted on by the same torque, the object with the larger moment of inertia has the smaller acceleration.
The moment of inertia of an object depends on the mass of the object, and on how this mass is distributed with respect to the axis of rotation. The farther the bulk of the mass is from the axis of rotation, the greater is the rotational inertia (moment of inertia) of the object. The units of the moment of inertia are units of mass times distance squared, for example kgm2.
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Imagine two wheels with the same mass.
One is a solid wheel, the mass is evenly distributed throughout the
structure, while the other has most of the mass concentrated near
the rim.
The wheel with the mass near the rim has the greater moment of inertia. |
The moment of inertia is defined with respect to an
axis of rotation. For example, the moment of inertia of a circular disk
spinning about an axis through its center perpendicular to the plane of the disk
differs from the moment of inertia of a disk spinning about an axis through its
center in the plane of the disk.
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Players and angular acceleration
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If during the the one-on-one collision the force due to the interaction with the other player is applied near the center of mass of of the player, it produces no torque, and therefore no angular acceleration. Immediately after the collision the player has no angular velocity about his center of mass.
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If, however, the player is tackled in such a way that the interaction force is applied away from the center of mass and produces a torque, the player emerges from the collision with angular velocity and will have a hard time staying upright.
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Step through the examples below to observe collisions from which a player emerges with or without angular velocity.
Coaches often advise linemen to stay low. With their center of mass closer to the ground, an opposing player has to tackle them near their CM. The lineman is then less likely to rotate, fall over, and be moved out of the way. This technique is critical for a defensive lineman in defending his own goal in the "red" zone, the last 10 yards before the goal line.
After the collision the player is only acted on by gravity and the normal and frictional force due to his interaction with the ground, unless he immediately collides with one ore more other players. These forces can also produce torques about various points in his body. The player may be able to shift his center of mass in such a way as to minimize these torques. A player at rest with a wide foot stance is better able to do this that a player running at full speed with only one foot at a time contacting the ground.
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