Atwood's Machine- Calc Based & Differentials

[bollocks748]

Homework Statement

Consider the Atwood's machine of Lecture 8. We wish to use this machine to measure our local acceleration of gravity with an accuracy of 5% [i.e. (Delta g)/g = 0.05]. To begin, suppose we let the mass m_1 fall through a distance L.

3.1 Find an expression for the acceleration of gravity, g, in terms of m_1, m_2, L and t.

3.2 Now suppose we are able to measure time with an accuracy of (Delta t) = 0.1 s. Assuming that, for example, (Delta t)/t can be approximated by the differential dt/t, write the relationship between (Delta g)/g and (Delta t)/t. You can do this by starting with the derivative dg/dt determined from the equation in the previous part.

3.3 If L = 3 m and m_1 = 1 kg, determine the value of m_2 required to determine g to 5%. If we want to measure g to 1% would the mass m_2 increase or decrease - why? (On your own, you might want to consider the effect of the uncertainty in the masses of m_1 and m_2 on the determination of g.)

The Attempt at a Solution

Okay, I got excellent help on one problem I struggled with, so hopefully I'll get some help on this one. Solving for the net force, I ended up with the equation that the acceleration downward is g * (m1-m2/m1+m2). Setting that equal to another expression for acceleration, L=1/2 a t^2, I end up with the function g= [2L(m1+m2)]/[t^2(m1-m2)]. Part 1 done.

Then for part two, I derived dg/dt and formed the differential equation:

dg= (-4L(m1+m2)/t^2(m1-m2))(dt)

Since I need dg/g and dt/t, I divided both sides by g, and added a t to the right side:

dg/g= [(-4*L*t*(m1+m2))/(g*t^3(m1-m2))](dt/t), which would finish part two.

Part three is where I run into an issue. I used 9.8 m/s^2 to fill in for g in the equation, and .1s for t, in order to solve for m2. However, I don't think that's the right approach, as I'm supposed to be measuring the local gravity, which may or may not be exactly 9.8 m/s^2. As well as that, it doesn't make sense to me that I can set dt and t to .1s each. But, by doing that approach, I reached and answer of 1.000817kg for m2. That is so close to m1's value that it doesn't seem correct either. Am I approaching differentials incorrectly? Thanks in advance to anyone who can help me out with this.

 

[bollocks748]

Just didn't want to get bumped to page 2. :-)

 

[baba89]

I would like to point out that Atwood's machine is a classic experiment used to measure the local acceleration of gravity. It consists of two masses connected by a string passing over a pulley, and the difference in their weights causes the system to accelerate. By measuring the acceleration of the system, we can calculate the local value of gravity.

In this problem, we are asked to find a way to measure the local acceleration of gravity with an accuracy of 5%. This can be achieved by using the differential equation dg/g= [(-4*L*t*(m1+m2))/(g*t^3(m1-m2))](dt/t) derived in part 2. This equation relates the relative error in the measurement of gravity (dg/g) to the relative error in the measurement of time (dt/t).

To solve part 3, we need to consider the values of L, m1, and t that are given. Substituting L=3m, m1=1kg, and t=0.1s, we get dg/g= -4(m1+m2)/3(m1-m2) * dt/t. To achieve an accuracy of 5%, we need to set dg/g=0.05. Solving for m2, we get m2=0.952kg. This is the value of m2 required to determine g to 5% accuracy.

To measure g to 1% accuracy, we would need to decrease the relative error in the measurement of gravity. This can be achieved by increasing the relative error in the measurement of time, which can be done by increasing the value of t. In other words, we need to measure the time with less precision in order to measure g with higher precision. This may seem counterintuitive, but it is a result of the relationship between the relative errors in the measurements.

In conclusion, the Atwood's machine is a useful tool for measuring the local acceleration of gravity with a high degree of accuracy. By understanding the relationship between the relative errors in the measurements, we can determine the necessary parameters to achieve a desired level of precision.