Accelerating system with hanging mass and friction lab

[chillpenguin]

Homework Statement

I am working on the analysis portion of a lab. The experiment was dropping varying masses connected to a cart by a string and pulley. The cart's mass was constant in each trial, but the hanging mass was increased each trial. We were measuring the acceleration of the cart. Graphing our acceleration data vs. mass1 / (mass1 + mass2) gives us a linear looking line. Using a curve fit in graphical analysis program, we can see the slope is around 9.0 m/s/s. This slope relates to gravity. However it is not actually g because of the friction that is slowing the system down. Now here is the question I am stuck on: Find an equation that you can use to graph your existing experimental data to estimate g and f (force of friction). Regraph your data using this equation and use linear fit to estimate g and f. I am stuck here. The idea is that accounting for friction gives a more accurate estimation of g (gravity). But I don't know what to do. I realize I could solve for f by assuming g = 9.8 but that is not the point. I need to get g through graphing my data somehow.

Homework Equations

The Attempt at a Solution

I figured out that I can expand on a = m1(g) / (m1+m2) by adding friction. I got:
a = (m1(g) - f) / (m1+m2)

where a = acceleration
m1= hanging mass
m2 = cart on track
f = force of friction

I'm trying to see how I can use this to work out an equation or a way to graph my data but I don't see how. I thought maybe acceleration could be y-axis, m1/(m1+m2) could be x, gravity could be slope (m), and y-intercept (b) would be the acceleration loss due to friction. But I'm not sure how to graph my data and get the results I want. How am I supposed to do this?

 

[chillpenguin]

Here is a nice image that demonstrates the 2 mass system.
http://clas.sa.ucsb.edu/staff/alex/physics6a/challengeproblems_files/image007.gif

 

[TSny]

What if you graph (m₁ + m₂)a on the y-axis? What could you then choose for the variable on the x-axis that would make a nice plot?

 

[chillpenguin]

I can't see what would give me g and f. I tried graphing (m1 + m2)a vs. m1 but that's the same thing I already did. I must be missing something.

 

[chillpenguin]

Was I right to assume that whatever equation I use, g will be the slope and f will be the y-intercept?

 

[TSny]

Was I right to assume that whatever equation I use, g will be the slope and f will be the y-intercept?

If you plot a vs. m₁/(m₁ + m₂), the graph would not be a straight line unless the friction force, f, is zero. But it would approximately be a straight line if f is small compared to m₁g. The y-intercept of the graph would not be equal to the friction force.

A straight line is of the form y = Mx + b where M and b are constants. In your case, you chose y to correspond to the acceleration, a, and x to correspond to the quantity m₁/(m₁ + m₂). But that doesn't give you an equation of the form
y = Mx + b where b is a constant.

 

[chillpenguin]

Well the part where I graphed y to correspond to the acceleration, a, and x to correspond to the quantity m1/(m1 + m2) was for estimating g without taking into consideration friction. g was the slope of that line. My lab instructions asked me to do that, and the b value was so small it was negligible. Now the question I am stuck on is asking me to improve upon my estimation of g by including f into the equation somehow and using a linear fit to get an estimation of both g and f. The frictional force is definitely very small because the hanging mass is pulling a cart with wheels on a track. So when I graphed a vs. m1/(m1 + m2) the line was straight. But my g value was not 9.8, it was 9.0. So I can only assume my g value will be closer to 9.8 if I set up a graph that accounts for friction somehow. But I do not know how to do that. My question is what do I need to graph to get g and f? The only data I have is 5 trials for m1, a, and a constant in all trials m2. Btw m2 is the cart and m1 is the hanging mass. Also I forgot to mention that f (friction force) is assumed to be a constant through this experiment.

 

[TSny]

What if you graph the quantity $(m_1+m_2)a$ on the $y$ axis?
What does this quantity equal in terms of $g$ and $f$?
What could you graph on the $x$-axis so that a graph of $y$ vs. $x$ would yield a straight line if $f$ is constant?