2004 Spring Electromagnetism

1. [33 points total] The figure below shows an elevator at rest in a gravitational field of strength such that objects fall freely with acceleration g. Inside of the elevator there is a balance with two arms of equal length. An iron cube (density $\rho_{Fe}$) of mass $M_{Fe}$ is hanging from the left arm by string of negligible mass and is submerged in water (density $\rho_{H_2O}$). On the other side of the balance there is an object of mass $M_c$. You may neglect the effects of atmosphereic pressure and surface tension.

A. [5 points] What is the total force exerted by the liquid on the iron cube?

Greg’s Solution:

The force is equal to the amount of water displaced by the cube,

\begin{align} F = \rho_{H_2O} V_{H_2O} g = \frac{\rho_{H_2O}}{\rho_{Fe}} M_{Fe} g \end{align}

B. [12 points] Give $M_c$ in terms of $M_{Fe},\rho_{Fe},\rho_{H_2O}$ for the balance to be in equilibrium.

Greg’s Solution:

The gravitational force acting on $M_c$ is just $M_c g$. We need for this to be equal to the gravitational force on the iron mass minus the buoyant force pushing it up, that is

\begin{align} M_c g = g M_{Fe} - \frac{\rho_{H_2O}}{\rho_{Fe}} M_{Fe} g \quad \Rightarrow \quad M_c = M_{Fe} - \frac{\rho_{H_2O}}{\rho_{Fe}} M_{Fe} \end{align}

C. [16 points] The elevator accelerates downward in the gravitational field with an acceleration $a=g/3$. How does your answer to part A change, if at all?

By the equivalence principle, this is equivalent to changing the amount of gravity felt by the system. Since the gravitational constant does not explicitly appear anywhere in the above equation, we conclude that accelerating the system does not alter the result.

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