An engineer is calibrating a device that uses a vertical ideal spring of unknown…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Force and Translational Dynamics · Force and Translational Dynamics

Problem

An engineer is calibrating a device that uses a vertical ideal spring of unknown force constant

k

to apply controlled forces to small test samples. By design, the spring will be compressed downward so that it exerts an upward force of

F_{\text{target}}

on the sample. A series of trial measurements yields the following pairs

(F_{\text{target}},\;|\Delta x|)

for the same spring: •

(4.0\,\text{N},\;0.85\,\text{cm})

•

(9.0\,\text{N},\;1.89\,\text{cm})

Assume the data are exact and that the spring obeys ideal behavior. (a) Derive an expression for

|\Delta x|

as a function of the desired target force

F_{\text{target}}

using the two data points provided. Show all reasoning steps, including determination of the spring constant

k

. (b) Using your expression from part (a), calculate the compression required to apply a force of

15.0\,\text{N}

to the sample. Express your answer in millimeters. (c) Suppose the maximum allowable compression for mechanical safety is

3.0\,\text{cm}

. Assess whether the spring can safely reach the

15.0\,\text{N}

target force. Explain your reasoning quantitatively.