Force of spring equation problems

[Elementard]

Homework Statement

A light spring having a force constant of 145 N/m is used to pull a 6.00 kg sled on a horizontal frictionless ice rink. The sled has an acceleration of 1.80 m/s2.

By how much does the spring stretch if it pulls on the sled horizontally?

Relevant Equations

Fs= -kx
Fp= ma

m=6.00Kg
K=145N/m
a=1.80m/s

Fp=Fs
ma=-kx
(6)(1.8)=-(145)x
x=-0.0745m

I’m just wondering why I see some people make Fs=kx instead of -kx? isn’t the force of a spring a vector?

 

[kuruman]

It is a vector. Say the sled is being pulled by the spring to the right. The spring stretches to the left which means that the displacement of the end of the spring is negative. One can formally write it as a one-dimensional vector in the negative direction using unit vector notation $\mathbf{x}=-|\mathbf{x}|~\mathbf{\hat x}$. Then one can write a vector equation for the force, $$\mathbf{F}=-k\mathbf{x}=-k(-|\mathbf{x}|~\mathbf{\hat x})=+k|\mathbf{x}|~\mathbf{\hat x}.$$ The negative sign in the one-dimensional vector equation, $F=-kx$ indicates the direction of $F$ relative to $x$. In this example with the spring to the right of the sled, if vector $x$ is to the left (spring extension) , vector $F$ is to the right; if vector $x$ is to the right (spring compression), vector $F$ is to the left.

 

[Elementard]

Thank you so much! This actually makes sense.

 

[kuruman]

I just noticed that your answer is a negative number. You have solved for the displacement of the end of the spring which is a vector. This is incorrect. The problem is asking you by how much the spring stretches. This is a distance, i.e. the magnitude of the displacement, and a positive number.

 

[haruspex]

I just noticed that your answer is a negative number. You have solved for the displacement of the end of the spring which is a vector. This is incorrect. The problem is asking you by how much the spring stretches. This is a distance, i.e. the magnitude of the displacement, and a positive number.

I would put it a little differently. "Extension" could be thought of as a distance or as a displacement.

Fs=-kx assumes the force being exerted by the spring at end A and the displacement of end A are being measured with positive in the same direction.
In this case, we are being asked for the extension, which is in the opposite direction to the force. Therefore the required answer is -x.