Can Atwood's Machine Measure Gravitational Acceleration with 5% Accuracy?

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Homework Statement

We hope to use Atwood's machine to measure gravitational accleration with an accuracy of 5% [(Delta g)/g = 0.05]. Mass m1 is heavier than m2 and it falls through a distance L.

We are able to measure time with accuracy of (Delta t) = 0.1s. If we say, for any x, (delta x)/x can be expressed as the differential dx/x, write the relationship between (Delta g)/g and (Delta t)/t.

Homework Equations

The Attempt at a Solution

Suppose T as a tension in the rope.

from
m1g - T = m1a
T - m2g = m2a,

g = (m1+m2)g/(m1-m2).

From L = 1/2at^2, a = 2L/t^2.


Therefore, g = 2L(m1+m2)/[(m1-m2)t^2]


I can't go further from this step...

any idea to solve this problem??

 

[anathema]

I understand your desire to measure gravitational acceleration with a high level of accuracy. To achieve this, we need to consider the sources of error in our measurements and how they affect our overall accuracy.

First, let's define our variables:
- (Delta g)/g represents the relative error in our measurement of gravitational acceleration
- (Delta t)/t represents the relative error in our measurement of time

Using your equations, we can express (Delta g)/g as:

(Delta g)/g = [(Delta m1)/m1 + (Delta m2)/m2 + (Delta t)/t]

This takes into account the errors in measuring the masses of the objects as well as the error in measuring time.

Now, let's consider the specific case of using Atwood's machine. We know that in this setup, the masses are falling through a distance L and we are measuring time with an accuracy of (Delta t) = 0.1s.

Substituting these values into our equation, we get:

(Delta g)/g = [(Delta m1)/m1 + (Delta m2)/m2 + 0.1/t]

Since we want an accuracy of 5%, we can set (Delta g)/g = 0.05 and solve for (Delta t)/t:

0.05 = [(Delta m1)/m1 + (Delta m2)/m2 + 0.1/t]

(Delta t)/t = 0.05 - [(Delta m1)/m1 + (Delta m2)/m2]

This tells us that in order to achieve a 5% accuracy in our measurement of gravitational acceleration, we need to have an error in our measurement of time that is less than 0.05 - [(Delta m1)/m1 + (Delta m2)/m2].

In summary, the relationship between (Delta g)/g and (Delta t)/t is dependent on the errors in measuring the masses of the objects as well as the time, and in order to achieve a certain level of accuracy in our measurement of gravitational acceleration, we must ensure that the error in our measurement of time is smaller than a certain value.

 

[baba89]

I can provide a response to this content by saying that Atwood's machine is a simple and effective tool for measuring gravitational acceleration. However, achieving an accuracy of 5% may be challenging due to various factors such as friction, air resistance, and human error. It is important to carefully design and conduct the experiment to minimize these sources of error.

The relationship between (Delta g)/g and (Delta t)/t can be expressed using the differential dx/x as follows:

(Delta g)/g = (2L/m1-m2)(Delta t)/t^2

This shows that the accuracy of the measurement of gravitational acceleration (Delta g)/g is directly proportional to the accuracy of the measurement of time (Delta t)/t^2. Therefore, in order to achieve an accuracy of 5% for (Delta g)/g, the accuracy of (Delta t)/t^2 should be at least 5% divided by the constant 2L/(m1-m2). This highlights the importance of accurate time measurements in determining gravitational acceleration using Atwood's machine.

Additionally, it is important to note that the accuracy of the masses m1 and m2 can also affect the overall accuracy of the measurement. Any errors in measuring the masses will also contribute to the overall error in the measurement of gravitational acceleration. Therefore, it is crucial to use precise and calibrated equipment for measuring the masses.

In conclusion, while Atwood's machine is a useful tool for measuring gravitational acceleration, achieving an accuracy of 5% may be challenging and requires careful consideration of various factors such as friction, air resistance, human error, and accurate measurements of time and masses.