(III) The pendulum in a grandfather clock is made of brass and keeps perfect time at 17^∘ C How

(III) The pendulum in a grandfather clock is made of brass...

Key Concepts

Simple Pendulum Dynamics

A simple pendulum’s behavior is governed by the formula T = 2??(L/g), where T is the time period, L is the length of the pendulum, and g is the gravitational acceleration. This relationship shows that the time period increases as the length of the pendulum increases, making pendulum clocks sensitive to any changes in length.

Thermal Expansion

Thermal expansion refers to the phenomenon where the dimensions of a material change in response to temperature variations. As a material heats up, its length typically increases, which can affect the accuracy of devices that depend on precise dimensions like pendulum clocks.

Coefficient of Linear Expansion

The coefficient of linear expansion is a property that quantifies how much a material's length increases per unit increase in temperature. This coefficient is essential for calculating the change in the pendulum’s length due to temperature fluctuations, which in turn affects its timekeeping.

Timekeeping Accuracy in Pendulum Clocks

Pendulum clocks rely on a precise period to keep accurate time. Any inadvertent changes in pendulum length, such as those caused by thermal expansion, directly alter the period of oscillation. This sensitivity underscores the importance of accounting for environmental factors in precision timekeeping.

Transcript

00:01 The pendulum in a grandfather clock is made of brass, and it keeps perfect time at 17 degrees celsius.

00:07 How much time is gained or lost in a year if the clock is kept at 28 degrees celsius? the frequency dependent on length or a simple pendulum applies.

00:17 So this time dependency on length is given by tau for time, because not to confuse it with key for temperature, is equal to 2 pi root l over g, where g is a gravitational constant, l is the length of a pendulum.

00:35 So we aren't given the length in the question.

00:37 So we need to make length the subject of this equation.

00:41 So we have length is equal to g over 4 pi squared, tau plus delta, tau, sorry, tau, sorry, is 4, is g over 4 pi tau squared.

01:04 And so if we perturb this equation here and make, give length a delta out, a small increase in the length because of the small temperature increase, then we expect to see a small increase in the period that it takes.

01:22 So length plus delta l is equal to g over 4 pi squared tau plus del tau tau squared so this just represents the the length of a pendulum for a given time period with a small increase in this case and we are going to call this equation one and this equation and equation three will be our formula for linear thermal expansion because we have a pendulum, a wire, basically a brass wire that's expanding as a function of time.

02:11 So we get that delta l is equal to alpha l delta t, where delta t is changing in temperature and not to be confused with tau.

02:27 So, and we're going to call this one equation three, and we're going to substitute in equation three into equation two here.

02:38 So doing that substitution, we get that l brackets 1 plus alpha delta t is equal to g over 4 pi squared, tau plus delta, all squared.

03:06 And we're going to call this one equation four.

03:10 And we're going to start a new page because this one's getting quite full.

03:15 So we're going to go from our next step is to substitute equation one into equation four.

03:22 And we're going to do that one over leaf to give ourselves a bit more space.

03:26 So doing that, we get that g and selecting g over 4 pi squared, tau squared.

03:42 So substituting just to double check, so g over 4 pi squared, tau squared, and then the rest of that.

03:47 So 1 plus alpha delta t, 1 plus alpha delta t is equal to, just double checking this.

04:00 Off is equal to g over 4 pi squared, tau plus delta, tau squared.

04:09 So g over 4 pi squared, tau plus delta tau squared.

04:22 Oh, no, the square needs to go outside.

04:29 So squared.

04:31 And we observed that we can divide both sides by g over 4 pi.

04:35 Squared to eliminate that term.

04:37 And we're also going to do a tailor expansion here.

04:42 Sorry, we're just going to expand the brackets here.

04:45 And we're going to observe that our change of time in our pendulum compared to a year is going to be very small.

04:51 So a term that's delta tau squared will be zero or will be close enough to zero to justify removing it...

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