∙∙∙A clock uses a pendulum that is 75 cm long. The clock is accidentally broken, and when it is

∙∙∙A clock uses a pendulum that is 75 cm long. The clock is...

$\bullet\bullet\bullet$ A clock uses a pendulum that is $75 \mathrm{~cm}$ long. The clock is accidentally broken, and when it is repaired, the length of the pendulum is shortened by $2.0 \mathrm{~mm} .$ Consider the pendulum to be a simple pendulum. (a) Will the repaired clock gain or lose time? (b) By how much will the time indicated by the repaired clock differ from the correct time (taken to be the time determined by the original pendulum in $24 \mathrm{~h}$ )? (c) If the pendulum rod were metal, would the surrounding temperature make a difference in the timekeeping of the clock? Explain.

Key Concepts

Simple Pendulum Dynamics

This concept involves understanding the motion of a simple pendulum where a mass is suspended from a fixed point and swings under the force of gravity. Its dynamics are described by a restoring force proportional to the sine of the displacement angle, and for small angles, the motion approximates simple harmonic motion with a period that depends on the length of the pendulum and the acceleration due to gravity.

Pendulum Period and Its Dependence on Length

The period of a simple pendulum is given by the formula T = 2??(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This relationship shows that the period is directly proportional to the square root of the pendulum's length, meaning that a decrease in length results in a decrease in the period, causing the pendulum to oscillate more quickly.

Effect of Length Changes on Timekeeping

In clocks that use a pendulum to measure time, any alteration in the pendulum's length directly affects the clock's accuracy. A shortened pendulum decreases the period, which causes the clock to run fast, meaning the time it indicates will be ahead of the actual time. The small fractional change in length leads to a proportional change in the period, and over many oscillations, this discrepancy accumulates into a noticeable timing error.

Thermal Expansion and Its Impact on Clock Accuracy

Thermal expansion refers to the change in a material's dimensions in response to temperature changes. In the context of a pendulum clock, if the pendulum rod is made of metal, its length can vary with temperature fluctuations. Such changes can alter the period of the pendulum, leading to timekeeping errors. Materials with different coefficients of thermal expansion affect the clock’s precision, necessitating compensation or the use of temperature-insensitive materials in precision clocks.

Transcript

00:02 In the first part of this problem, we have to compare the period of the simple pendulum with the length of the pendulum.

00:09 Let's try the expression for the period of the simple pendulum.

00:13 As t is equals to 2 pi square root of l divided by g.

00:18 Here, this l is the length of the pendulum and this g is the gravitational acceleration.

00:28 Since the t is proportional to the square root of l, which is the length of the pendulum, therefore shortening the length will decrease the period and consequently the panellation.

00:37 Pendulum will lose that time by taking the fast vibrations.

00:43 Now let's move to the part b of this problem.

00:46 So in part b, we have to find the difference between the time measured by the two pendulums.

00:54 So we need to calculate this delta t.

01:01 Now let's calculate the period of the unbroken pendulum.

01:06 So for unbroken pendulum, we can wide the period as t1 is equals to 2 pi square root of the length which is equal to 75 multiplied by 10x per minus 2 meter divided by the gravitational acceleration on the surface of earth that is equals to 9 .81 meter per second square so this will give us the value for this t1 as 1 .7 37 seconds now the number of vibrations taken by the by this pendulum in the 24 hours is written as so this is the number of vibrations.

01:56 It should be equal to the time taken by spondelum to complete this number of vibrations divided by the period of this pendulum which is t1.

02:04 So this can be written as 24 hours multiply by 3 ,600 seconds per hour divided by this period which is equal to 1 .737 seconds.

02:19 So from here we can wide the number of vibrations as n is equal to equals to 49 ,741.

02:32 So these are the number of vibrations taken by this pendulum...

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