Chapter 2 – Excerpt 4
Another short excerpt, with a high concentration of feline toys and habits, including a cartoon
Applications of Torque
For example, if you are sitting on a seesaw that is in an exactly horizontal position, then the gravitational force on you is perpendicular to the radial vector, which is along the board, so the magnitude of the torque is just Fd. Suppose you weigh 10 pounds and your person’s daughter weighs 30 pounds and the two of you are perfectly balanced on a seesaw. To keep it from rotating, you must be sitting exactly 3 times as far away from the fulcrum at the center as she is sitting. If you are sitting any closer than that, the seesaw will tilt so that you are higher up. Such a moment has been captured in the sculpture shown in Figure 2-10.
A given force can be made more effective at producing torque by increasing either the length of the lever arm or the sine of the angle. The former is the theory behind long-handled tools, such as wrenches, hammers and axes. Holding them at the end of the handle while twisting or swinging them is most effective. And of course it is also important to direct the tool’s motion at a right angle to the handle, maximizing the value of sinθ. A contrasting situation is illustrated in Figure 2-11, where the longer the fork the greater the effort required to hold the piece of meat up, to balance the torque exerted by gravity.
Q. Two cats are playing with a tennis ball of radius 8 cm. Simultaneously, the smaller cat applies a force of 1 newton at the top edge of the ball while the larger one pushes it in the opposite direction three times as hard at a point 3 cm. above the floor. See Figure 2-12. Describe the motion of the ball around an axis through the center of the ball parallel to the floor and perpendicular to the page. Which cat had the greater effect on the ball’s spin?
A. The torque due to the smaller cat is rF = (8cm.)(2 N) = .16 Nm., since the force is perpendicular to the radial vector. For the larger cat’s force, the angle from the radial vector to the force vector is the supplement of θ, which has the same sine of (r-3)/r = 5/8. So that torque is (8 cm.)(3 N)(5/8)= .15 Nm. Thus the smaller cat has a slightly larger effect on the ball’s spin and the ball will rotate clockwise around the given axis.
Moment of Inertia
An object rotating or revolving about an axis tends to keep rotating or revolving about that axis, unless there is a change to the force. Note that unlike Newton’s First law, this is not a statement about what happens in the absence of a force. It’s about what continues to happen in the presence of the same continual force.
We saw in Chapter 1 that mass, or inertia, is a measure of an object’s tendency to resist changes to its motion. The resistance to changes in rotational motion possessed by an object is called its rotational inertia, or moment of inertia. It is symbolized by I. (Note that the uppercase ninth letter of the Greek alphabet, iota, appears identical to the uppercase ninth letter of our alphabet.)
As an aside, just to break up the monotony of this discussion, consider the misnomer “cat nap”: Although humans know that cats nap extensively, they use the phrase “cat nap” to mean a brief period of rest. Perhaps “moment of inertia” would be a better term for this phenomenon. See Figure 2-13.
The rotational inertia of an object depends not only on its mass but also on the distribution of that mass. The further a piece of mass is from the axis of rotation, the more it contributes to the rotational inertia. This makes intuitive sense because it takes more torque to get it to revolve through the same angular displacement.
We would like the rotational analog of Newton’s Second Law to hold. Since the analogs of F,m and a are, respectively, τ, I, and α, the analog of F = ma would be
τ = I α.
Let’s determine what the rotational inertia I would have to be in order to make this true. Using a = rα and τ = rF yields I = τ/α = rF/(a/r) = (F/a)r2 = mr2. Thus a point mass m at a distance r has a rotational inertia of mr2.
The moment of inertia is proportional to mass, and proportional to the square of distance. Therefore, if the mass of, say, a thin hoop is doubled, so is its moment of inertia; but if its radius is doubled, then its moment of inertia is quadrupled.
4 – The item pictured is actually a piece of tofu., not only to keep this book kosher but also to keep the handler safe from an overeager attack. <return-to-text>
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