Question 5.8: Weighing a Fish in an Elevator A person weighs a fish of mas...
Weighing a Fish in an Elevator
A person weighs a fish of mass m on a spring scale attached to the ceiling of an elevator as illustrated in Figure 5.14.
(A) Show that if the elevator accelerates either upward or downward, the spring scale gives a reading that is different from the weight of the fish.
(B) Evaluate the scale readings for a 40.0-N fish if the elevator moves with an acceleration a_y=\pm 2.00 m/s^2.
(A) Conceptualize The reading on the scale is related to the extension of the spring in the scale, which is related to the force on the end of the spring as in Figure 5.2. Imagine that the fish is hanging on a string attached to the end of the spring. In this case, the magnitude of the force exerted on the spring is equal to the tension T in the string. Therefore, we are looking for T. The force \overrightarrow{T} pulls down on the spring and pulls up on the fish.
Categorize We can categorize this problem by identifying the fish as a particle in equilibrium if the elevator is not accelerating or as a particle under a net force if the elevator is accelerating.
Analyze Inspect the diagrams of the forces acting on the fish in Figure 5.14 and notice that the external forces acting on the fish are the downward gravitational force \overrightarrow{F}_g= m\overrightarrow{g} and the force \overrightarrow{T} exerted by the string. If the elevator is either at rest or moving at constant velocity, the fish is a particle in equilibrium, so \Sigma F_y=T-F_g=0 or T=F_g=m g. (Remember that the scalar mg is the weight of the fish.)
Now suppose the elevator is moving with an acceleration \overrightarrow{a} relative to an observer standing outside the elevator in an inertial frame. The fish is now a particle under a net force.
Apply Newton’s second law to the fish:
\Sigma F_y=T-m g=m a_ySolve for T:
(1) T=m a_y+m g=m g\left(\frac{a_y}{g}+1\right)=F_g\left(\frac{a_y}{g}+1\right)
where we have chosen upward as the positive y direction. We conclude from Equation (1) that the scale reading T is greater than the fish’s weight mg if \overrightarrow{a} is upward, so a_y is positive (Fig. 5.14a), and that the reading is less than mg if \overrightarrow{a} is downward, so a_y is negative (Fig. 5.14b).
(B) Evaluate the scale reading from Equation (1) if \overrightarrow{a} is upward:
T=(40.0 N)\left(\frac{2.00 m/s^2}{9.80 m/s^2}+1\right)=48.2 NEvaluate the scale reading from Equation (1) if \overrightarrow{a} is downward:
T=(40.0 N)\left(\frac{-2.00 m/s^2}{9.80 m/s^2}+1\right)=31.8 NFinalize Take this advice: if you buy a fish by weight in an elevator, make sure the fish is weighed while the elevator is either at rest or accelerating downward! Furthermore, notice that from the information given here, one cannot determine the direction of the velocity of the elevator.
