F=ma 2014 solutions
P01
P01 solution
P02
P02 solution
P03
P03 solution
P04
P04 solution
P05
P05 solution
P06
P06 solution
P07
P07 solution
P08
P08 solution
P09
P09 solution
P10
P10 solution
P11
P11 solution
P12
The following information applies to questions 12 and 13
A paper helicopter with rotor radius \(r\) and weight \(W\) is dropped from a height \(h\) in air with a density of \(\rho\).
Assuming that the helicopter quickly reaches terminal velocity, a function for the time of flight \(T\) can be found in the form
\[ T = k h^\alpha r^\beta \rho^\delta W^\omega\]
where \(k\) is an unknown dimensionless constant (actually, 1.164). The exponents \(\alpha\), \(\beta\), \(\delta\), and \(\omega\) are constants to be determined.
Determine \(\alpha\).
(A) \(\alpha = -1\)
(B) \(\alpha = -\frac{1}{2}\)
(C) \(\alpha = 0\)
(D) \(\alpha = \frac{1}{2}\)
(E) \(\alpha = 1\)
P12 solution
Since the helicopter quickly reaches terminal velocity \(v_t\), from kinematics, the time \(T\) of flight is given by
\[ T = \frac{h}{v_t} \propto h^1\]
Thus, \(\alpha = 1\) so the answer is \(\boxed{E}\).
P13
Determine \(\beta\).
(A) \(\beta = \frac{1}{3}\)
(B) \(\beta = \frac{1}{2}\)
(C) \(\beta = \frac{2}{3}\)
(D) \(\beta = 1\)
(E) \(\beta\) cannot be uniquely determined without more information.
P13 solution
From the previous problem, we found \(T = k h^\alpha r^\beta \rho^\delta W^\omega\). Taking the dimensions of both sides,
\[ [T] = [h]^\alpha [r]^\beta [\rho]^\delta [W]^\omega\]
Using the fact that \([\rho] = \frac{M}{L^3}\) and \([W] = \frac{ML}{T^2}\),
\[ T = L^1 L^\beta \left( \frac{M}{L^3} \right)^\delta \left( \frac{ML}{T^2} \right)^\omega\]
Counting powers of \(T\):
\[ 1 = -2\omega \\ \omega = -\frac{1}{2}\]
Counting powers of \(M\):
\[ 0 = \delta + \omega \\ \delta = -\omega = \frac{1}{2}\]
Counting powers of \(L\):
\[ 0 = 1 + \beta - 3\delta + \omega \\ \beta = 3\delta - \omega - 1 = 4\omega - 1 = 2 - 1 = 1\]
Since \(\beta = 1\), the answer is \(\boxed{D}\).