Mass — Question 5

Question 5

An unknown resistance can bias a one-point mass estimate

In two positive-motion tests, a cart’s applied force P and acceleration a are $(5\ \mathrm N,1\ \mathrm{m/s^2})$ and $(9\ \mathrm N,3\ \mathrm{m/s^2})$. An unknown constant resistance R opposes motion, so $P-R=ma$.

1. Find m and R.

2. What mass would the naive ratio $P/a$ give for each test?

3. Explain the advantage and limitation of the two-test method.

Question 5 - Solution

An unknown resistance can bias a one-point mass estimate

(a) Subtract the equations.

$$9-5=m(3-1)\ \Rightarrow\ \boxed{m=2\,\mathrm{kg}},\quad R=5-2(1)=3\,\mathrm N$$ $9-5=m(3-1)$ gives $m=2$ kg. Then $R=5-2(1)=3$ N.

(b) Expose the bias.

$$m_{\mathrm{naive,1}}=5/1=5\,\mathrm{kg},\quad m_{\mathrm{naive,2}}=9/3=3\,\mathrm{kg}$$ The naive masses are 5 kg and 3 kg, neither the correct 2 kg. The applied force is not the net force.

(c) State the shared-offset assumption.

$$\Delta P=m\Delta a+\Delta R,\quad\frac{\Delta P}{\Delta a}=m\quad\text{only if }\Delta R=0$$ Subtraction removes a resistance that is the same in both tests. If resistance changes with speed or conditions, two observations do not justify this fixed-R fit without more evidence.