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Unformatted text preview: mass m lying on a horizontal, frictionless
turntable is connected to a string attached to the center of
the turntable, as shown in Figure 6.14. According to an inertial observer, if the block rotates uniformly, it undergoes an
acceleration of magnitude v 2/r, where v is its linear speed.
The inertial observer concludes that this centripetal acceleration is provided by the force T exerted by the string and
writes Newton’s second law as T mv 2/r. n Noninertial observer n T Fﬁctitious = mg T mv 2
r mg (a) Inertial observer (b) Figure 6.14 A block of mass m connected to a string tied to the center of a rotating turntable.
(a) The inertial observer claims that the force causing the circular motion is provided by the force T
exerted by the string on the block. (b) The noninertial observer claims that the block is not accelerating, and therefore she introduces a ﬁctitious force of magnitude mv 2/r that acts outward and balances
the force T. Optional Section 6.4
4.9 163 MOTION IN THE PRESENCE OF RESISTIVE FORCES In the preceding chapter we described the force of kinetic friction exerted on an
object moving on some surface. We completely ignored any interaction between
the object and the medium through which it moves. Now let us consider the effect
of that medium, which can be either a liquid or a gas. The medium exerts a resistive force R on the object moving through it. Some examples are the air resistance associated with moving vehicles (sometimes called air drag ) and the viscous
forces that act on objects moving through a liquid. The magnitude of R depends
on such factors as the speed of the object, and the direction of R is always opposite
the direction of motion of the object relative to the medium. The magnitude of R
nearly always increases with increasing speed.
The magnitude of the resistive force can depend on speed in a complex way,
and here we consider only two situations. In the ﬁrst situation, we assume the resistive force is proportional to the speed of the moving object; this assumption is
valid for objects falling slowly through a liquid and for very small objects, such as
dust particles, moving through air. In the second situation, we assume a resistive
force that is proportional to the square of the speed of the moving object; large
objects, such as a skydiver moving through air in free fall, experience such a force. 164 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws
a=g v R vt v 0.63vt mg (a) t τ v = vt
a=0 (c) (b) Figure 6.15 (a) A small sphere falling through a liquid. (b) Motion diagram of the sphere as it
falls. (c) Speed – time graph for the sphere. The sphere reaches a maximum, or terminal, speed
vt , and the time constant is the time it takes to reach 0.63vt . Resistive Force Proportional to Object Speed
If we assume that the resistive force acting on an object moving through a liquid
or gas is proportional to the object’s speed, then the magnitude of the resistive
force can be expressed as...
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This document was uploaded on 09/19/2013.
- Fall '13
- Circular Motion