1. period

A pendulum swings back and forth 10 times in 8s. It swings through a total horizontal distance of 40cm.
What is the period? I know it is 8s/10 times but I am not convinced. I think it is 10 times / 8 s = 1.25 times/s . Can some one explain to me why? thank you

Also when you sketch the graph, is the range from -20cm to 20cm or 0cm to 40 cm. I think it is 0cm to 40 cm because distance can not be negative, however in the text book in is the other range.

2. Re: period

You've got period and frequency confused (easy to do since they're closely related). Period describes how long it takes for the pendulum to complete one full cycle, while frequency describes how many cycles the pendulum will complete in a given time. Period is measured in seconds, frequency in cycles/second (aka hertz).

frequency = 10 cycles / 8 seconds = 1.25 cycles/second
period = 8 seconds / 10 cycles = 0.8 seconds (the cycles disappear from the unit because period is defined as "the time to complete one cycle")

Dimensional analysis is your friend with physics word problems. If you aren't sure whether to multiply or divide, go with the one that makes the units come out correct. For example, if a problem gives you a distance and a time, and asks for velocity, and you know that velocity = distance/time, regardless of how confusing the problem is there's only one way to put the numbers together to make the units come out right.

As for the graph, distance can't be negative, but displacement can. At rest, the pendulum hangs straight down (zero displacement), you pull it to the right (positive displacement) and let it go. It swings back to its starting point (zero displacement) and keeps going (into negative displacement). The cycle repeats. It all depends on what you choose as your reference point (zero). Usually physics problems choose the "at rest" state as the origin of the graph. By sketching the graph from 0cm to 40cm you're putting the origin at the point where the pendulum is just about to swing back the other way. Given the wording of the problem, your graph isn't wrong, though a teacher might take off points because it doesn't follow the convention.

3. Re: period

Thank you very much! This really cleared up a few points.

Edit: Btw this was a math word problem. Also if the period is 0.8s , this represents the x-axis . However we are using radians in this unit. So does that mean 0.8s = 0.8 radians?