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Hollow or solid cylinderFig. 38 shows a hollow cylinder of length The moment of inertia is given by Volume Hence and the moment of inertia is
If the cylinder is solid If the cylinder is very thin,
Note, that the moment of inertia of cylinder does not depend on the length
5.3.3. Uniform sphere of radius Divide the sphere into thin disks. The radius Its volume is and its mass is Hence from Eq. () Integrating this expression from 0 to Carrying out the integration, we obtain The mass Hence
Torque A doorknob is located as far as possible from the door's hinge line for a good reason. If you want to open a heavy door, you must certainly apply a force; that alone, however, is not enough. Where you apply that force and in what direction you push are also important. If you apply your force nearer to the hinge line than the knob, or at any angle other than 90° to the plane of the door, you must use a greater force to move the door than if you apply the force at the knob and perpendicular to the door's plane. Figure 11-15a shows a cross section of a body that is free to rotate about an axis passing through Î and perpendicular to the cross section. A force To determine how
The ability of equivalent ways of computing the torque are And
called the line of action of Torque, which comes from the Latin word meaning "to twist," may be loosely identified as the turning or twisting action of the force
Torque and work, however, are quite different quantities and must not be confused. Work is often expressed in joules (1 J = IN- m), but torque never is. In the next chapter we shall discuss torque in a general way as being a vector quantity. Here, however, because we consider only rotation around a single axis, we do not need vector notation. Instead, a torque has either a positive or negative value depending on the direction of rotation it would give a body initially at rest: If the body would rotate counterclockwise, the torque is positive. If the object would rotate clockwise, the torque is negative. (The phrase "clocks are negative" from Section 11-2 still works.) Torques obey the superposition principle that we discussed in Chapter 5 for forces: When several torques act on a body, the net torque (or resultant torque) is the sum of the individual torques. The symbol for net torque is 11-9 Newton's Second Law for Rotation A torque can cause rotation of a rigid body, as when you use a torque to rotate a door. Here we want to relate the net torque
where Proof of Equation 11-34 We prove Eq. 11-34 by first considering the simple situation shown in Fig. 11-16.
The rigid body there consists of a particle of mass A force
The torque acting on the particle is, from Eq. 11-32,
From Eq. 11-22 (
The quantity in parentheses on the right side of Eq. 11-35 is the rotational inertia of the particle about the rotation axis (see Eq. 11-26). Thus, Eq. 11-35 reduces to For the situation in which more than one force is applied to the particle, we can generalize Eq. 11-26 as which we set out to prove. We can extend this equation to any rigid body rotating about a fixed axis, because any such body can always be analyzed as an assembly of single particles.
Date: 2015-01-12; view: 1595
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