Describing rotational motion

Newton's Laws for rotational motion

1. An object that is rotating tends to remain rotating at the same rate around the same axis
   The First Law for rotation accounts for the gyroscope effect (which stabilizes a spinning object like a football or frisbee or rolling objects such as bicycle tires or bowling balls), and the tendency of a spinning skater, diver, or gymnast to keep spinning (the hard part is not to keep spinning, but to stop gracefully!).

   All rotating objects have "rotational inertia" that tends to keep them rotating, however some objects have more than others. The "angular momentum" is proportional to the "moment of inertia" which depends on the mass of an object and how it is distributed. The distribution matters because in a rotation object the points farther from the axis of rotation are moving faster than points closer in. If there is a lot of mass far from the axis, then that object will have a larger fraction of its mass moving fast and a larger angular momentum than something with its mass concentrated at the axis of rotation. For a given mass, a longer stick-like object will have a larger moment of inertia than a shorter one (the exact relationship is I=m*L²/12, where L is the length and m is the mass). If instead all the mass was concentrated at the two ends, the moment would be I=m*L²/4. The moment of inertia also depends on the location of the axis of rotation. The above formulas assumed an axis of rotation through the center of the stick and perpendicular to the long axis of the stick. If you held the stick at the end and swung it, then it would have I=m*L²/3 (larger than either value above). If you had all the mass at one end and swung it from the other, then you would have a much larger value: I=m*L².
2. In Newton's Second Law, when there is an imbalance of forces, the velocity will change at a rate inversely proportional to the mass:

   Dv/Dt = F/m

   In rotational motion, Whenever there is an imbalance of angular forces (know as torques, with a symbol: t), then there will be a change in the angular velocity, the rate of change in the angular velocity is proportional to the net torque and inversely proportional to the moment of inertia:

   Dw/Dt = t/I

   Two objects with the same mass and same size can have very different moments of inertia such as the two loaded sticks I passed around in class. One stick had weights close to the center and was easy to rotate (low I) but the other stick was weighted at each end and was more difficult to rotate (high I). For a rotating object, the points on the object far from the axis of rotation are moving faster than the points closer to the axis of rotation, as a result, the more the mass is concentrated farther from the center, the higher the moment of inertia.
   Clearly a higher I would be better for something like a bat which when it strikes a much lighter object will not have its rotation slowed too much [the bat is weighted asymmetrically at the end opposite the handle, of course]. On the other hand, a low moment is what you want to execute fast rotations [gymnasts, divers, and skaters try to minimize their moments of inertia at such times].

   Torque

   So what is torque? Torque is what produces changes in the rotation of an object. Torque is produced by any force that acts on an object that is not pointed directly towards the object's axis of rotation (or for a freely rotating object, that would be its center of mass). Examples of torque in action:

   • The force of friction at an athlete's feet (the reaction force to the athlete pushing in the opposite way). In the diagram A below, the athlete pushed to the right and friction pushes back to the left, the resulting torque will cause the athlete to rotate clockwise. When executing a flip, the athlete always pushes down and either forward or back to initiate the rotation.
   • an impulse applied away from the axis of rotation will cause an abrupt change in rotational speed (see diagram B). This happens when a bat hits a ball, or a foot kicks a ball.
   • The couple: a combined pushing and pulling at different points along an extended object (see diagram C). If the two forces are equal and opposite the net force on the object is zero (so the center of mass does not accelerate).
   • whenever a struck (or bouncing) ball experiences a frictional force (the frictional force acts tangent to the ball's surface, perpendicular to the center of the ball)--the ball's spin will change (see diagram D).

   
   Examples of forces that don't produce torque:

   • gravity cannot produce torque around an axis at the center of mass (gravity acts as if it pulls on the object at the center of mass, so gravity is perfectly aligned with that axis); therefore gravity produces no torque on a flying object. [a diver or gymnast in the air can't change their angular momentum]
   • the normal force acting on a ball (the normal force is perpendicular to the surface, which is toward the center)
   • the normal force of the ground on a balanced object (which means that the center of mass is directly above the support point)

   The torque equals the strength of the force times the length of the "lever arm" (which is the minimum distance between the axis of rotation and line along which the force is directed):

   t = F*L

   For example, in swinging a bat, one can try to whip the bat around with the wrists or pull the bat around with the arms. Even if the wrists can exert a great force on the bat, the small lever arm compared to the radius of motion makes the force relatively ineffective in speeding up the bat. Instead, with the arms extended, the lever arm is moved out to the radius of motion, and the force is more efficiently applied.

   An impulse applied tangentiall at a large radius is very effective at changing the rotational rate, while an impulse applied at a shorter radius or less tangentially is less effective.

   If no torque is applied to an object, then it's "Rotational Momentum" (defined above) is constant. This means that the produce of I*w is constant. Even while spinning or twisting freely, skaters, gymnasts, and divers know they can control their angular speed by changing their moment of inertia. A skater speeds up the rate of spin by drawing the arms close to the body [divers who want to twist do the same]. A gymnast or diver who wants to flip goes into a tuck position to flip rapidly, then straightens the body to slow the rate of rotation.
3. if one object exerts a torque on another object, then that object exerts and equal and opposite torque back
   A good example of the Third Law for rotation occurs approaching the contact point in a golf or baseball swing: the increase in the rotation rate of the club or bat coincides with a sudden decrease in the rotation rate of the swinger's body around the same axis (in fact, this is where the power comes from, not the arms, although strong arms are needed to transfer the power from the body to the club). Another good example of the Third Law: swinging the arms when running is good because the motion of one leg forward while the other leg goes back tends to twist the body (which wastes energy since it does nothing to help the runner move forward) the motion of the arms in the opposite direction helps to counteract the twisting of the body.

   Generally, we use the rotational versions of Newton's Laws the way we use Newton's Laws in understanding translational motion: to help us to learn about forces acting on an object: if we see that an object begins to spin (or stops spinning) then we must figure out what force is acting and how it is acting to produce that change in rotation.