Chapter 2 – Excerpt 2
The next excerpt from Chapter 2 is fairly mathematical. It introduces the concepts of centripetal force and torque.
Whenever an object is revolving at constant linear speed v in a circle of radius r, its linear acceleration a has magnitude
and direction toward the center of the circle. If you’d prefer to placidly accept this expression rather than see how it’s derived, just skip the following box.
With the center of the circle as the origin, the displacement vectors are radial, and the tangential velocity vector is always perpendicular to the displacement vector. Δs is the change in the displacement vector (colored yellow) and Δv is the corresponding change in the velocity vector (colored purple). Both triangles shown are isosceles triangles with the same angle θ between their equal sides, so they are similar triangles. Hence we have Δv/v = Δs/r or Δv = v/r Δs. Since this is true for all Δs and Δv, we have a = dv/dt = lim Δv/Δt = lim (v/r Δs) /Δt = v/r lim Δs/Δt = v/r v = v2/r. It should be clear that in the limit as Δv goes to 0, the (aqua and magenta) velocity vectors approach each other and their difference, Δv, becomes perpendicular to them and thus points toward the center of the circle.
Note that although angular velocity is actually a vector, we have so far only been considering its signed magnitude, the angular speed. For example, equation (2) does not say that v = rω but rather that |v| = r|ω|. This is because the direction of the angular velocity, as well as the direction of the angular displacement, is very non-intuitively taken, by convention, to be perpendicular to the plane of the motion, along the axis of rotation. When we discuss forces, we will examine this issue more.
Since rotational motion is not uniform in direction, we know, by Newton’s First Law, that a force is needed to maintain it. Such a force is called a centripetal force. “Centripetal” means “center-seeking” in Latin. A force in the opposite direction is called “centrifugal”, which means center-fleeing.
These adjectives describe the direction of a force, not its source or nature. Any force that is continually perpendicular to the instantaneous direction of motion is a centripetal force. If it’s the only force present, it causes circular motion of its object, with the center of the circle being the fixed point toward which the force is directed. Thus the gravitational pull of the earth on the moon, which causes the moon to revolve around the earth, is a centripetal force, as is the force exerted by the rope on a tether ball in play, or by a whirling elastic string on an attached toy mouse.
Imagine that the toy mouse’s string were to suddenly break (since it is a more likely occurrence than the earth’s gravitational field suddenly disappearing, and can be made even more likely if you claw at it). Then the mouse would fly off on a tangent. This motion would not be due to a centrifugal force but rather to the absence of the centripetal force. Without the string’s pull on the mouse, the toy would continue to move at a uniform speed in a straight line, in the direction it was instantaneously going at the moment of breakage.
If you have ever watched your person make pizza, perhaps you have seen him repeatedly toss the dough into the air, spinning it. He was not doing this just to tantalize you or to show off, nor was it due to the same instinct that inspires your pre-prandial spinning tosses of a dying mouse. The purpose of the spinning tosses of pizza dough is to evenly spread the dough out into a thinner crust. This spreading out appears to be due to a centrifugal force, but again, the outward motion is an attempt by each piece of dough to continue on a tangential path, which is being combated by the pull of the more inner pieces of dough on it.
As in the above examples, it is generally the case that what appears to be a centrifugal force is actually the unexpected absence of a centripetal force, so centrifugal forces are sometimes called fictitious forces. However, in the frame of reference of someone in the revolving object, such as a bug trapped inside a tether ball, the centrifugal force appears real.
You may have experienced this trapped phenomenon while being driven to the vet. If the car taking you to the vet is rounding a curve toward the left, you will be flung toward the right side of the car. Within the frame of reference of the car, you would tend to feel as if a centrifugal force were acting on you, but from out on the road it’s clear that the road’s centripetal force acting on the car is merely not acting on your body.
Similarly, in Zepto Chan’s favorite toy (see Figure 2-6), the outer wall exerts a centripetal force on the revolving ball, forcing it to stay on its circular path.
Whenever a point mass m is revolving at speed v in a circle or radius r, the centripetal force on it that sustains this motion has magnitude
F = mv2/r.
This follows from applying F=ma to equation (3). We saw in the derivation of that equation that the direction of acceleration was radially inward; this is now supported by our intuition that the force is toward the center of the circle.
Although a force could be acting in any direction, it is only the component of the force in the direction perpendicular to the motion that sustains circular motion. Thus we can say that a centripetal force is the perpendicular component of a force. Now let’s consider the effect of the rest of the force – i.e. the tangential component of the force. (See Figure 2-7(a).) It is an obvious trigonometric fact that the magnitude of this component is Fsinθ , where θ is the angle between the force and the radial line. This tangential component is the component that causes changes to the circular motion.
You have probably noticed that when you try to push open a door, the closer you push to the side of the door where the knob is located, the easier it is to accomplish opening the door. In other words, the further your push is from the hinges, i.e. from the axis of the door’s rotation, the more effective your force is at causing the door to rotate. That, of course, is why the knob is located on that side of the door. It is also why cat doors should be located on the other side – so that if the human door happens to be slightly ajar when you go through the cat door, it won’t disturb you by moving if you happen to brush against a side of the cat door.
If you’re extremely observant as well as experimentally inclined, you may even have noticed that when you push twice as far from the axis of rotation, you only have to push half as hard to cause the same rotational motion of the door. Regardless, the observation that the effect of the tangential force on rotation varies directly with the distance from the axis of rotation hopefully leads to the conjecture that the dependency is linear, i.e.. that the effect is proportional to the distance from the axis to the point where the force is applied. We’ll call this radial distance r. It’s the magnitude of the radial position vector, which points from the axis of rotation to the point where the force is being applied.
Thus we have the conjecture that the rotation depends on the quantity
Indeed it does, and this quantity is actually the magnitude of a vector called the torque, or the moment of the force.
Torque is symbolized by the Greek letter τ (tau). In SI units, it is measured in newton-meters (Nm). Although these are the same units as those of joules, which measure work and energy, torque is not a form of energy. Note that the distance being considered here is perpendicular to the direction of motion, whereas for work the distance is along the direction of motion.
Torque influences rotational motion analogous to the way force influences translational motion. In the absence of torque, there is no change to rotational motion. Note that the absence of torque does not mean the absence of force: A force being applied at the center does not constitute torque because r = 0 for it; and a force being applied parallel to the motion arm does not constitute torque because sin θ = 0 for it (in fact, that’s the centripetal force that is sustaining the rotation).
In translational mechanics, the natural state of motion is one of constant velocity, which may be zero at rest, and any changes to it require a force. In rotational mechanics, the natural state of motion is rotation at a constant angular velocity, and changes to it require a torque.
The amount of torque is the radial distance r times the tangential component of the force, the component of the force perpendicular to the radial vector. Another way to view it is as the magnitude of the force times the component of the radial vector perpendicular to the line of action of the force. (See Figure 2-7(b).)) This distance that is the perpendicular component of the radial vector is called the moment arm or the lever arm. The latter term is due to the fact that simple lever motion is based on this principle. Thus the magnitude of the torque is the magnitude of the force times the moment arm. Using both momentous terms, we can say the moment of the force is the product of the moment arm and the force.
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