Pendulum clock error with thermal expansion and faulty barometer

[fasterthanjoao]

this is my second set of questions today, revising for an exam just now. would be grateful to anyone that can answer

1) The pendulum of a clock consists of a thin steel rod carrying a small heavy bob. the clock keeps correct time at 15 degrees centigrade, calculate the error in twenty four hours if the temperature rises to an average value of 20 degrees C.

coefficient of linear thermal expansion of steel = 11x10^-6

2) The reading of a true barometer and one containing a little air are 760mm and 700mm. when the air is partly exhausted from the space surrounding them they read 380mm and 360mm. Find the height from the mercury level in the reservoir to the top of the faulty tube.


thanks again, hopefully I won't have to post any-more problems later

 

[cepheid]

1. I hope that this question is asking you to find the error in the next twenty-four hours after the temperature reaches 20 degrees. Because if it is asking you to find the error as the temperature rises to 20 degress over the course of twenty-four hours, then that sounds much more difficult to me.

Let's assume the former scenario is what was intended. So, what happens to the steel when the temperature rises? It will expand, and you know exactly by how much. You've been given the coefficient of thermal expansion, which, if I remember right, tells you how much the length of the rod changes, as a fraction of the original length, per degree Celsius.

So, knowing by how much the length increases, what can you say about the period of oscillation of the pendulum? How does the length of a pendulum affect its period (what equation shows the relationship explicitly)? How will the change in period affect its timekeeping? From that, you can calculate the error that will accumulate in 24 hrs.

 

[wais]

1) To calculate the error in 24 hours, we can use the formula: error = (coefficient of linear thermal expansion) x (change in temperature) x (length of pendulum).
First, we need to convert the temperature change from degrees Celsius to degrees Kelvin by adding 273.15. So the change in temperature is 5.15 degrees K. The length of the pendulum is not specified, so we will use a general value of 1 meter. Plugging in these values, we get:
error = (11x10^-6) x (5.15) x (1) = 0.00005665 meters.
Therefore, the error in 24 hours is approximately 0.00005665 meters or 0.05665 millimeters.

2) To find the height, we can use the formula: h = (difference in mercury levels) x (total height of barometer) / (difference in barometer readings).
Plugging in the values, we get:
h = (760mm - 700mm) x (total height) / (380mm - 360mm)
Simplifying, we get:
h = 60mm x (total height) / 20mm
h = 3 x (total height)
Since the faulty tube is half filled with air, the total height of the barometer is twice the height from the mercury level to the top of the tube. Therefore, the height from the mercury level to the top of the faulty tube is 3/2 of the total height.
So, h = 3/2 x (total height)
This means that the height from the mercury level to the top of the faulty tube is 3/2 or 1.5 times the height from the mercury level to the top of the barometer.
Therefore, the height from the mercury level to the top of the faulty tube is 1.5 x (760mm - 700mm) = 1.5 x 60mm = 90mm.

 

[wais]

1) To calculate the error in twenty four hours, we can use the formula:

Error = (coefficient of linear thermal expansion) x (length of pendulum) x (change in temperature)

Substituting the given values, we get:

Error = (11x10^-6) x (length of pendulum) x (20-15) = 0.000055 x (length of pendulum)

To calculate the length of the pendulum, we can use the formula:

T = 2π√(L/g)

Where T is the time period, L is the length of the pendulum and g is the acceleration due to gravity.

At 15 degrees C, the time period is correct, so we can substitute the values to find the length of the pendulum:

2π√(L/9.8) = 2

√(L/9.8) = 1/π

L/9.8 = 1/π^2

L = 9.8/π^2

Substituting this value in the error formula, we get:

Error = 0.000055 x (9.8/π^2) = 0.000055 x 0.0983 = 0.0000054 meters

Therefore, the error in twenty four hours would be 0.0000054 meters.

2) To find the height from the mercury level in the reservoir to the top of the faulty tube, we can use the formula:

P1V1 = P2V2

Where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.

Initially, the pressure is 760mm and the volume is the height from the mercury level in the reservoir to the top of the tube. When the air is partly exhausted, the pressure becomes 380mm and the volume is the same as before.

Substituting the values, we get:

760mm x (height) = 380mm x (height)

Solving for height, we get:

Height = 380mm/760mm = 0.5 (since the units cancel out)

Therefore, the height from the mercury level in the reservoir to the top of the faulty tube is 0.5mm.