Dimensional Analysis in Halliday/Resnick

[Blargian]

Hello world! I'm busy working through Halliday/Resnick Fundamentals of Physics chapter 1 and I'm having some difficulty with this particular question. I'll do my best to explain my reasoning and attempt at the problem

1. Homework Statement

The problem: Three different clocks A, B and C run at different rates and do not have simultaneous readings of zero. Figure 1-6 shows simultaneous readings on pairs of the clocks for four occasions. (At the earliest occasion, for example, B reads 25.0 s and C reads 92.0s.) If two events are 600 s apart on clock A, how far apart are they on (a) clock B and (b) clock C? (c) When clock A reads 400 s, what does clock B read? (d) When clock C reads 15.0 s, what does clock B read? (Assume negative readings for prezero times.)

See diagram below:

https://www.dropbox.com/s/nihu76a6o7xwskv/physicsdiagram.jpg?dl=0

https://www.dropbox.com/s/nihu76a6o7xwskv/physicsdiagram.jpg

Homework Equations

This is more of a logic problem than anything else. The question states that A, B and C do not have simultaneous readings of zero), in other words the clocks were not started at the same time. Because the rates are different one would need to find the ratios of the clock speeds in relation to each other.

The Attempt at a Solution

A) We see that for A and B the interval readings line up, although at different rates. In order to compare we need to make sure we are comparing over the same interval so we subtract 312 from 512 for A and 125 from 290 for B. We can then get the ratios A/B and B/A. A is 1.21 x B and B is 0.825 x A. Therefore if two events are 600s apart on A they are (600)(0.825) = 495 seconds on B. This matches the book's answer.

B) Since we have already obtained the ratio of A to B, we now just need B to C. We subtract 25.0 from 200 for B and 92.0 from 142 for C to give ratios of 175/50 and 50/175 or 3.5 : 0.286... therefore 600 on A (495 on B) is equal to 495(0.286...) = 141s when rounded off. This is the textbooks answer.

C) This is where i go wrong. I said if Clock A = 400 then using the ratio obtained in question 1 B will be 400(0.825) = 330s. This is wrong and the answer should be 198s

D) I said if C=15.0s then B would read 52.5s as the ratio of C to B is 0.286: 3.5. The correct answer is -245s.

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The solutions manual states that the time on any of the clocks is a straight-line function of that on another, where slopes cannot = 1 and y-intercepts cannot = 0. From this they deduce

tC = 2/7(tB) + 594/7 and tB = 33/40(tA) - 662/5

This makes sense to me that each time would be a function of another. the gradient of each function is the ratio to the next however I do not understand how the Y-intercept of the functions (the + 594/7 and -662/5) where obtained from the graph.

Any explanation of this would be greatly appreciated.
- Shaun

 

[DrClaude]

C) This is where i go wrong. I said if Clock A = 400 then using the ratio obtained in question 1 B will be 400(0.825) = 330s. This is wrong and the answer should be 198s

I hope that you realized yourself that this was the wrong answer, as 400 s on the A clock will necessarily be between 125 s and 290 s on the B clock.

I do not understand how the Y-intercept of the functions (the + 594/7 and -662/5) where obtained from the graph.

Once you have found the ratio between two clocks, you take one point where simultaneous readings are made, and figure out what is the offset between the clocks.

 

[Orodruin]

C) This is where i go wrong. I said if Clock A = 400 then using the ratio obtained in question 1 B will be 400(0.825) = 330s. This is wrong and the answer should be 198s

D) I said if C=15.0s then B would read 52.5s as the ratio of C to B is 0.286: 3.5. The correct answer is -245s.

You are here assuming that the clocks showed zero at the same time, exactly what the problem told you not to do.

 

[Blargian]

I hope that you realized yourself that this was the wrong answer, as 400 s on the A clock will necessarily be between 125 s and 290 s on the B clock.

I didn't when i was working the problem. But i see now this is the case due to the fact that the clocks were started at different times.

Once you have found the ratio between two clocks, you take one point where simultaneous readings are made, and figure out what is the offset between the clocks.

Thanks very much for your reply! I took a point on A where they lined up and 'converted it' into B time using the ratio and then subtracted the two values to obtain the offset. Same for B and C.