# A spring scale A and a lever scale B having

A spring scale A and a lever scale B having equal lever arms are fastened to the roof of an elevator, and identical packages are attached to the scales as shown. Knowing that when the elevator moves downward with an acceleration of 1 m/$s^2$ the spring scale indicates a load of 60 N, determine (a) the weight of the packages, (b) the load indicated by the spring scale and the mass needed to balance the lever scale when the elevator moves upward with an acceleration of 1 m/$s^2$. Image from: F. P. Beer, R. E. Johnston, and P. J. Cornwell, Vector mechanics for engineers: Dynamics, 11th ed. New York, NY, United States: McGraw-Hill Inc.,US, 2015.

#### Solution:

Let us draw a free body diagram showing the forces affecting the package. We will use the following equations to solve this problem (Newton’s second law).

$F=ma$

$W=mg$

(Where $F$ is force, $W$ is weight, $m$ is mass, $a$ is acceleration, and $g$ is acceleration due to gravity)

a)

$+\uparrow\sum F=ma$

$F_s-W=-ma$

(Here, $F_s$ is the spring force, and $ma$ is negative because it’s pointing downwards, just as $W$, which is also pointing downwards)

Substitute $m=\dfrac{W}{g}$:

$F_s-W=-\dfrac{W}{g}(a)$

$F_s=-\dfrac{W}{g}(a)+W$

(factor out W)

$F_s=W(-\dfrac{a}{g}+1)$

$W=\dfrac{F_s}{(-\dfrac{a}{g}+1)}$

Substitute the values we know:

$W=\dfrac{60}{(-\dfrac{1}{9.81}+1)}$

$W=66.8$ N

b) Now that our acceleration is positive, we can again use the same process.

$W=\dfrac{F_s}{(\dfrac{a}{g}+1)}$

(Notice how our $\dfrac{a}{g}$ term is now positive because our acceleration is positive)

We know the value of $W$, and we are now isolating for $F_s$

$66.8=\dfrac{F_s}{(\dfrac{1}{9.81}+1)}$

$F_s=(66.8)(\dfrac{1}{9.81}+1)$

$F_s=73.6$ N

To figure out the mass needed to balance the scale, remember that on the scale, both sides must have equal forces (otherwise, it wouldn’t balance out). Knowing this and knowing that $W=mg$, we can write:

$m=\dfrac{W}{g}$

$m=\dfrac{66.8}{9.81}$

$m=6.81$ kg

(In essence, we simply found the mass that balances out the weight. The weight is mass times acceleration due to gravity, so to find the mass, we divide the weight by the acceleration due to gravity).

a) $W=66.8$ N
b) $F_s=73.6$ N
$m=6.81$ kg