Topic: Other
Concepts: Dimensional analysis

Solution:

The dimensions of $g$ are

so it is necessary (though not sufficient) to have equipment that can measure quantities with length dimensions and quantities with time dimensions.

• A) The mass $M$ can be hung on the spring scale which reads force $F=Mg$ so $g=F/M$.

• B) The mass can be dropped while we measure the time $T$ it takes the mass to travel across the length $L$ of the rod. Then since
  
  $L=\frac{1}{2}gT^2$
  
  we have $g=2L/T^2$.

• C) None of the equipment measure quantities which contain dimensions of length, so it is not possible to measure $g$.

• D) The mass $M$ can be launched upward with velocity $v$ and the height $h$ reached can be measured with the meter stick. Then since
  
  $\frac{1}{2}Mv^2=Mgh$
  
  we have $g=v^2/2h$.

• E) The motor of power $P$ can act on the mass $M$ for time $T$, accelerating it to speed $v$. By energy conservation,
  
  $PT=\frac{1}{2}Mv^2$
  
  $v=\sqrt{\frac{2PT}{M}}$
  
  We then measure the time $T_2$ it takes the mass to move across the string and determine the string length via $L=vT_2$. Knowing $L$, we can use the method described in choice B to measure $g$.

Thus, the answer is C.