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Chapter 12: Q8CP (page 497)
At room temperature, show that . It is useful to memorize this result, because it tells a lot about what phenomena are likely to occur at room temperature.
The result can be proved using the energy equation
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A box contains a uniform disk of mass M and radius R that is pivoted on a low-friction axle through its centre (Figure 12.58). A block of mass m is pressed against the disk by a spring, so that the block acts like a brake, making the disk hard to turn. The box and the spring have negligible mass. A string is wrapped around the disk (out of the way of the brake) and passes through a hole in the box. A force of constant magnitude F acts on the end of the string. The motion takes place in outer space. At time the speed of the box is , and the rotationalspeed of the disk is . At time the box has moved a distance x, and the end of the string has moved a longer distance d, as shown.
(a) At time , what is the speed of the box? (b) During this process, the brake exerts a tangential friction force of magnitude f. At time , what is the angular speed of the disk? (c) At time , assume that you know (from part b) the rotational speed of the disk. From time to time , what is the increase in thermal energy of the apparatus? (d) Suppose that the increase in thermal energy in part (c) is . The disk and brake are made of iron, and their total mass is . At time their temperature was . At time , what is their approximate temperature?
Consider the exponential function . Evaluate this function for x = 1, 10,000, and 0.01.
Explain why it is a disadvantage for some purposes that the specific heat of all materials decreases a low temperature.
In Chapter 4 you determined the stiffness of the interatomic “spring” (chemical bond) between atoms in a block of lead to be 5 N/m, based on the value of Young’s modulus for lead. Since in our model each atom is connected to two springs, each half the length of the interatomic bond, the effective “interatomic spring stiffness” for an oscillator is
4 × 5 N/m = 20 N/m. The mass of one mole of lead is 207 g (0.207 kg). What is the energy, in joules, of one quantum of energy for an atomic oscillator in a block of lead?
Suppose that the entropy of a certain substance (not anEinstein solid) is given by , where ais a constant. Whatis the specific heat capacity Cas a function of the temperature T?
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