Temperature Change in a Clock. A pendulum clock is designed...

Temperature Change in a Clock. A pendulum clock is designed to tick off one second on each side-to-side swing of the pendulum (two ticks per complete period). (a) Will a pendulum clock gain time in hot weather and lose it in cold, or the reverse? Explain your reasoning. (b) A particular pendulum clock keeps correct time at $20.0^{\circ} \mathrm{C}$ . The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to $10.0^{\circ} \mathrm{C} ?$ (c) How many seconds per day will the clock gain or lose at $10.0^{\circ} \mathrm{C} ?$ (d) How closely must the temperature be controlled if the clock is not to gain or lose more than 1.00 $\mathrm{s}$ a day? Does the answer depend on the period of the pendulum?

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of a material’s dimensions to change when its temperature changes. In solids, this behavior is quantified by the coefficient of linear expansion, which describes how much a unit length of material expands or contracts per degree change in temperature. This concept is critical in understanding how structural elements, like the pendulum’s shaft in a clock, respond to temperature variations.

Pendulum Period Dependence on Length

The period of a simple pendulum, which is the time taken for one complete swing, is directly related to the square root of its length (T = 2??(L/g)). This means that even small changes in the length of the pendulum, due to thermal expansion or contraction, can cause noticeable changes in its period. Understanding this relationship is essential for analyzing how a clock's timekeeping is affected by temperature changes.

Fractional Changes and Error Propagation

When the length of a pendulum changes by a small fractional amount due to a temperature shift, the period of the pendulum also changes by a fraction that is approximately half of the fractional change in length. This propagation of error from the physical dimension to the timing mechanism is a key factor in calculating how much a clock might gain or lose time over a period due to temperature fluctuations.

Temperature Sensitivity in Timing Devices

Temperature sensitivity in timing devices involves assessing how environmental temperature changes affect the accuracy of devices such as pendulum clocks. This includes determining acceptable limits for temperature variation to keep timekeeping errors within a desired threshold, and understanding whether these effects are intrinsic to the physical properties of the timing mechanism or can be compensated for through design adjustments.

Transcript

00:01 For this problem on the topic of temperature and heat, we are told that a pendulum clock, which is designed to take off one second on each side -to -side swing of the pendulum, is subject to a temperature change, and we want to know if the pendulum will gain time in hot weather and lose time in cold weather or the reverse.

00:19 We're then told that the clock keeps the correct time at 20 degrees celsius, and the pendulum shaft is made from steel.

00:26 We want to know the fractional change in the length of the shaft when it is cooled by a 10 degrees celsius and how many seconds per day will be lost or gained at this loss of 10 degrees celsius.

00:38 And lastly, we want to know how closely the temperature needs to be controlled so that the clock cannot gain or lose more than a second a day.

00:48 Now, in hot weather, the moment of inertia i and the length d for the equation of the period will both increase by the same factor.

00:57 And so the period will be longer and the clock will run slow.

01:01 So in hot weather, weather and increase in the period means that the clock will lose time.

01:09 Similarly, the clock will do the same and will run fast in cold weather since the period will be shorter and so the clock gains time in cold weather.

01:32 So in part b, we are told that a particular pendulum clock keeps the correct time at 20 degrees celsius, which is made from steel, and we want to know the fractional change in its length when it is cooled by 10 degrees celsius.

01:45 So the fractional change in its length is delta l over l0, and this is equal to alpha delta t.

01:59 Now we know alpha, the linear expansion coefficient for steel is 1 .2 times 10 to the minus 5 per celsius degree times the change in temperature of 10 degrees celsius, or 10 celsius degrees degrees.

02:24 And so we get the ratio of the changing length to the original length to be 1 .2 times 10 to the minus 4.

02:41 So that's the fraction by which the length of the clock will change by a decrease of 10 degrees celsius...

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