## 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.
Mathematical Derivation
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 = v 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 Centripetal Force 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 These adjectives describe the 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 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 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 = mv 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. Torque
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 – 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, If you’re extremely observant as well as experimentally inclined, you may even have noticed that when you push 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 of the force.momentTorque 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 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 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 . 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.lever arm© All Contents Copyright 2008-2010, Skona Brittain |
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