How Does Simple Harmonic Motion Depend on Pendulum Length and Gravity?

[Pichi123]

Homework Statement

1. Modeling. Functions.
A student measures the length L = 3 feet of a light (0.1 grams) string and the mass m = 7
kilograms of a small (radius of 0.0005 inches) bob at the end of the string.
The students also knows the value of acceleration due to gravity g = 9.8
meters/second-squared at the location of the experiment.
(S)he constructs a simple pendulum, and sets it in motion of a small amplitude. (S)he uses a
stop-watch to measure the period T1 of the pendulum.
Then the students quadruples the length of the string and conducts the experiment again,
measuring the new period T2.

1)
a. What is the ratio T1/T2? Use dimensional analysis to answer this question. Hint: the
answer is provided in question #2. You do not have to justify 2\pi coefficient, just the
root.
b. If the student moves the experiment to the moon, will the period increase or decrease?
c. Discuss the approximations involved in the model if they are sufficient to assume
harmonic motion.
d. Assuming the constant g = 9.8 meters/second-squared, view T as a function of L. Set
up a limit to express instantaneous rate of change of T with L. What is the unit for this
rate?

2)
Functions. definition of a limit.
Your stop-watch is broken, and you can only measure L and compute the period from the
formula:

T= 2π\sqrt{L/g}

a. If L = 3 feet, what is the max error permitted in the length measurement so you are
still within 0.5 seconds of the “correct” answer for the period? Please, convert
everything to SI system of units, if needed. The SI system uses
"meter-kilogram-second" basic units.
b. What is the max persentage error permitted in the length measurement so you are still
within 1% of the “correct” answer for the period?

Homework Equations

T1/T2

\sqrt{L/g}

T= 2π\sqrt{L/g}

Lim T(L)
L-->?

The Attempt at a Solution

I was hoping I could have some of my answers/equations checked, and there are a few I'm not even sure what I'm supposed to do.

for 1a, my equation was \sqrt{L/g}/\sqrt{4L/g}, and my answer was 1/2.

for 1b, I put that the period increases

1c I wasn't sure how to answer...if it actually wanted me to discuss every approximation or simply state if the information given is enough to calculate simple harmonic motion.

in 1d I wasn't sure what the limit was...I'm thinking it's supposed to be 9.8 but I want to be sure. And from there, I'm not sure what to do.

in part 2, I'm not sure what to do in either one at all...you see, I'm doing this assignment for calculus in college, and I've completely forgotten my physics from high school. The professor never mentioned how to calculate the max error permitted, max percentage error, etc...

any and all help will be very much appreciated. thanks in advance

 

[vela]

Homework Statement

1. Modeling. Functions.
A student measures the length L = 3 feet of a light (0.1 grams) string and the mass m = 7
kilograms of a small (radius of 0.0005 inches) bob at the end of the string.
The students also knows the value of acceleration due to gravity g = 9.8
meters/second-squared at the location of the experiment.
(S)he constructs a simple pendulum, and sets it in motion of a small amplitude. (S)he uses a
stop-watch to measure the period T1 of the pendulum.
Then the students quadruples the length of the string and conducts the experiment again,
measuring the new period T2.

1)
a. What is the ratio T1/T2? Use dimensional analysis to answer this question. Hint: the
answer is provided in question #2. You do not have to justify 2\pi coefficient, just the
root.
b. If the student moves the experiment to the moon, will the period increase or decrease?
c. Discuss the approximations involved in the model if they are sufficient to assume
harmonic motion.
d. Assuming the constant g = 9.8 meters/second-squared, view T as a function of L. Set
up a limit to express instantaneous rate of change of T with L. What is the unit for this
rate?

2)
Functions. definition of a limit.
Your stop-watch is broken, and you can only measure L and compute the period from the
formula:

T= 2π\sqrt{L/g}

a. If L = 3 feet, what is the max error permitted in the length measurement so you are
still within 0.5 seconds of the “correct” answer for the period? Please, convert
everything to SI system of units, if needed. The SI system uses
"meter-kilogram-second" basic units.
b. What is the max persentage error permitted in the length measurement so you are still
within 1% of the “correct” answer for the period?

Homework Equations

T1/T2

\sqrt{L/g}

T= 2π\sqrt{L/g}

Lim T(L)
L-->?

The Attempt at a Solution

I was hoping I could have some of my answers/equations checked, and there are a few I'm not even sure what I'm supposed to do.

for 1a, my equation was \sqrt{L/g}/\sqrt{4L/g}, and my answer was 1/2.

The question is worded kind of strangely. I believe you're supposed to use dimensional analysis to reason that the period T is proportional to the square root of L/g. That's why it says to not worry about the factor of 2π but justify the square root. Once you have that, you can figure out the ratio T₁/T₂ as you did.

for 1b, I put that the period increases

You should explain why you think so.

1c I wasn't sure how to answer...if it actually wanted me to discuss every approximation or simply state if the information given is enough to calculate simple harmonic motion.

A simple pendulum doesn't actually undergo simple harmonic motion. It approximately does when a certain condition is met. Go back over the analysis of the simple pendulum in your textbook, and you should see what approximation was used. Is the assumption on which the approximation based valid for your set-up? If not, modeling its motion as simple harmonic motion probably isn't very good.

You might also compare the idealized model to the actual physical set-up. The idealized model, for instance, assumes a massless string, but real strings have mass. Is approximating it as 0 reasonable? What effect does the string's mass have on the motion? And so on.

in 1d I wasn't sure what the limit was...I'm thinking it's supposed to be 9.8 but I want to be sure. And from there, I'm not sure what to do.

The instantaneous rate of change is the derivative dT/dL. Use the definition of the derivative to write that down as a limit.